π, the ratio of a circle’s circumference to its diameter, is an irrational number, which means it can’t be written as a fraction a/b, where a and b are integers. That means that, unlike decimals like 1/4, or 0.25, or repeating decimals, like 1/3 or 0.33333333, it neither terminates nor repeats. It goes on forever without repeating itself: 3.1415926. . . . ad infinitum.

Now you can prove that pi must be irrational; Wikipedia gives six explanations, and I’ll put one video proof below. All of these proofs depend not on the geometry of a circle, but on the fact that pi appears in certain trigonometric relationships.

What I think is weird is not pi’s irrationality itself, but simply the fact that a ratio that so important for geometry turns out to be an irrational number. Why couldn’t it be THREE? As Jason Rosenhouse pointed out a long time ago, the Bible implies that it is three, showing that God was ignorant. And you may be aware that at the end of the 19th century, the Indiana state legislature considered (but rejected) a bill including an assertion that the value of pi was 3.2.

The answer whyπ is irrational is, I suspect, simply “that’s the way it is.” But if there’s some proof out there that the ratio of a circle’s circumference to its diameter, based on the geometry alone, must be irrational, I’d like to know about it.

And I’m a bit surprised that nobody uses the irrationality of π as an inexplicable fact about mathematics that implies the existence of God. And then the astute theomathematician could bring up the square root of two. . .

“Almost all numbers are irrational so no surprises this one is.”

This is not actually true. There are infinitely many rational numbers, and infinitely many irrationals.

“Infact pi is also trancsendental which is also unsurprising as almost all irrationals are”

Again, not actually true. All transcendental numbers are irrational, but there are infinitely many irrational numbers that are not transcendental, such as the square root of 2.

There are countably many rationals but uncountably many irrationals (Google Cantor). In fact the cardinality of the irrationals is equal to the cardinality of the power set of the rationals. (If there’s an infinity with intermediate cardinality we’ll leave as an exercise for the reader…)

Sorry, but in terms of set theory, what you have said here is not correct. The cardinality of the irrational numbers is strictly larger than the cardinality of the rational numbers.

If you want a proof of this, cf. Cantor’s Diagonal Proof. Well, sort of…the proof deals with the cardinality of the real numbers and the integers. However, a bit more work will show that the rational numbers and the integers have the same cardinality, and that the irrational numbers and the real numbers have the same cardinality.

True. In terms of measurement theory (part of real analysis) the set of rational numbers within the set of real numbers has measure zero. In other words, the size of the set of real numbers and the size of the set of irrational numbers is exactly the same.

In fact, Pi goes beyond just being irrational. It is transcendental, that is, it is not the solution for any polynomial P(x) = 0, where the coefficients of P(x) are integers. Note that some irrational numbers, such are sqroot 2, are not transcendental (aka algebraic).

This fact is why the classic problem “squaring the circle” has no solution.

The ever famous e is also a transcendental number.

Oh…just noticed yossarian’s post. He mentions transcendental as well.

In addition to being irrational, pi is also “transcendental” meaning it is not the zero of any polynomial with rational coefficients. This is Not the case with square roots.

As Yossarian says above, pi is not only irrational but transcendental, meaning it is not the root of any polynomial with integer coefficients. Non-transcendental numbers are called “algebraic”. Sqrt(2) is an example of algebraic, irrational number.
The set of algebraic numbers is countable, meaning it has the same cardinality as the set of integers. You need to include the transcendental numbers to make the set of real numbers uncountable. This means that the vast majority of real numbers are transcendental. In fact, if you could somehow pick a real number at random your chances of getting a non-transcendental number would be zero.

The “chances” can be proven mathematically to be zero. To do it, you first have to define things. You enter the field of measure theory. For example, all the numbers between 0 and 1 have measure = 1, all the numbers between 1/4 and 1/2 have measure = 1/4. Defining the measure precisely requires a fair bit of formal math, the examples given are just what falls out of the formal definition. It can then be rigorously proved that the measure (hence probability) of a number being non-transcendental is 0.

A rough Google search suggests 39 digits – so to 38 decimal places – is enough to calculate the … volume? … of the known universe to within one hydrogen atom.

But that leaves lots of room for the subatomic particles I guess…

Yossarian – I never thought about that. Is there a number theory picture for that, akin to how primes are the building blocks for all other whole numbers? Is it a mistake to expect geometry to have some perfect relationship with numbers?

For more pi trivia check Neil DeGrasse Tyson’s pi day tw33ts.

You only need (roughly) as many decimal digits as there are orders of magnitude in your margin of error.

If you’re using a meterstick that’s one meter long and has centimeter markings, 3.1 is about as accurate a measurement as you’re going to make. If it’s marked in millimeters, 3.14 will do the trick.

If you remember your “Powers of 10” short film from the ’60s or so, you’ll recall that there’s only a couple dozen orders of magnitude in either direction between us and the extremes of bigness and smallness. So, yes, a few dozen digits of π is ample for all engineering at any scale, and an hundred digits is beyond meaningless for even the most speculative of theoretical physics.

(Though there might be a footnote for certain multiverse theories, including the simplest which gives universal repetititition on the scale of a googolplex lightyears or so.)

What really intrigues me about pi is that it can be calculated in several different ways that have nothing to do with geometry, let alone circles. That implies that it’s a number that has nothing to do with the properties of our universe. If there are parallel universes that have different laws of nature than ours, pi will still be the same number there that it is here.

Pi is so easily described using any number of very elegant and simple infinite series. Are these really weirder and more supernatural than 1/3 which is an infinite string of threes in base ten? http://www.geom.uiuc.edu/~huberty/math5337/groupe/expresspi.html?

Here is an interesting corollary. Imagine a circle, and imagine a string that is the length of the circle’s diameter. Place one end (let’s call it the left end) of the string somewhere on the circle, align the string to the circle, and note the location where the other end (let’s call it the right end) of the string is on the circle. Now move the string along the circle so that the left strand is now where the right strand used to be, and note the new position of the right string. Repeat the process as many times as you want, you will NEVER EVER end up on the same location. You can repeat the process an infinite number of time and each time the location at the end of the string on the circles will be different from any other locations found before. Cool way to illustrate infinity😉

If we use Pi = measured C/D, then in a curved space-time, Pi can be smaller. Take the sun for example: for a fixed orbital circumference, the radius will be larger than Newton would predict because the sun’s mass warps spacetime. Thus if you calculated Pi based on C/D of the sun, and you were really accurate, you would get number slightly smaller than its standard mathematical value.

But this has little to do with pi’s utility in helping to solve various problems. The easiest way to respond to the above point is to simply say ‘we don’t define pi based on an empirical measure, we define pi based on a mathematical relation…which in this case includes a flat geometry’

In curved geometries, the ratio of C/D is variable, but (barring singularities) converges on pi as D approaches zero. There is no geometry, curved or flat, in which the ratio of C/D is a constant other than pi.

There many hidden assumptions there, though, like a manifold structure etc. On a hexagonal lattice C/D converges to 3 as we reach the smallest size, and π as the size increases.

Still it’s an interesting fact for smooth cases – could you provide some references for a proof, please?

It might help to go back to the very basics of the meanings of the terms.

“Rational” simply means that there is an whole-number fraction, or ratio, that represents that value. You can cut one apple into three pieces; two of those pieces represent two thirds of an apple.

Were π rational, you could take a piece of string and fold in exactly so many fractions to mark off the perimeter of a circle, and fold it in some other number of fractions to mark off the diameter.

But it’s instantly obvious that you can’t do that for any small whole-number ratio, and it’s practically trivial to create other sorts of geometric objects which also don’t have any whole-number ratio relationships.

So the real question is why you should expect to be able to do this with circles.

Perhaps the better question then would be why you might expect any particular property to be expressible as whole-number ratio relationships. And, in that light, maybe the true surprise is that Pythagorean triangles are expressible rationally….

But it’s instantly obvious that you can’t do that for any small whole-number ratio,

To play devil’s advocate, I don’t think so. Let’s say I measured and cut a string corresponding to C, and then marked it with evenly spaced marks. Then I use that string to measure the diameter and find out that the point were the loose end of the string crosses the circumference doesn’t align with any mark on my string. My first thought (assuming I don’t know geometry) would *not* be “huh, I guess that means no mark will ever align, no matter how closely I space them together.” Instead, my thought would be “I need more finely divided marks to get the exact measure.” I’m not sure the answer that it’s irrational is instantly obvious, IOW. Its true, yes, but not instantly obvious based on human experience or perception. Which is why the bible probably gets it wrong; because its not instantly obvious.

But what reason do you have in the first place to expect that there should be such a relationship? And how far must you go on such a quixotic quest to question your premise?

Archimedes knew that 22/7 was close but not it, at which point you should suspect that there’s no such ratio. Ptolemy pushed it to 377/120, which is more than enough to cast overwhelming doubt. By the Middle Ages there were infinite series calculation methods, at which point it should become intuitively obvious that the ratio isn’t even remotely simple.

But what reason do you have in the first place to expect that there should be such a relationship?

I think in terms of human intuition, the idea that no matter how closely you drew the marks they would never line up with the circumference would be baffling and surprising. The intuitional expectation is that some mark will,you just have to find it.

But that’s just me, Ben. Maybe your own intuition naturally leads you to the right conclusion. Even if that were the case, though, I would be wary of proclaiming your own intuited result as “instantly obvious” to people in general. I think its a very common failing for people who understand something (especially some deductive or mathematical relation) to think that what they understand must be obvious to others. It isn’t. My own personal favorite was professor at Jerry’s Alma Mater, W&M, who declared to his students (including a friend of mine) that the QM wavefunction for a particle in a box was “intuitively obvious.” Do you think he was right, or do you think he possibly made a mistake in thinking something he knew like the back of his hand must be equally obvious to others?

If deductive, logical, and mathematical relationships were instantly obvious, we wouldn’t need math classes.

“Trivial” was the favorite word to one of my grad school classmates. Every proof he new how to do was trivial – except for once in a while when you asked him to demonstrate and he’d forgotten how.

I was more aiming towards the fact that we shouldn’t have any initial expectations in such cases — either for or against rationality. And, once you start investigating the matter, the evidence quickly piles up against rationality. The ancients already knew enough to be highly skeptical of suggestions of rationality, and it was really only superstition (aka “philosophy”) that caused them to cling to rationality long after reasonably supportable.

You’ll find many people decidedly unimpressed with philosophy, especially its central conceit that one can think one’s way to understanding reality.

It is perfectly possible to be an highly-respected philosopher and to advocate positions contradicted by observation, let alone ones not supported by observation. And it is depressingly common for philosophers to disdain the notion that the obvious way to answer many questions is to go out and make observations.

Those trolley car “thought” “experiments” are the perfect example. No industrial safety engineer would even pretend to find anything useful about any of that nonsense. Psychologists learned the relevant lessons with Milgram et al. and you’d have a damned hard time getting approval for such past an ethics review board. (Authority figure says to do something horrific and people do — we know that.) But philosophers just eat up that bullshit.

So long as you care more about how you think the world should be rather than go out and have a look at what it actually is, you’re doing philosophy. And it should be unsurprising that the people wearing philosopher’s hats who’re doing real, meaningful work are making or analyzing observations…which is what the rest of us know as…

…wait for it…

…science.

And it’s been that way since antiquity. Archimedes exclaimed, “Eureka!” after observing the relationship between density, weight, and volume. But Aristotle never bothered to check if things might not keep going if there’s nothing holding them back (what, not even any rumors of frozen-over lakes?), and so we got stuck with that whole catastrophe of metaphysics that still plagues us to this day.

I mean…damn, but that’s William Lane Craig’s whole problem in his debate with Sean Carrol. WLC couldn’t escape Aristotle’s trap if his life depended on it, when the rest of the world moved on literally centuries ago once Newton got bopped on the head by an apple.

How do we know which philosophical notions are and aren’t correct? Do we further philosophize about them, or do we go out and check the notions against the evidence?

If you can point to a single philosophical notion deserving of serious consideration that is not supported by equally-serious observation, I’ll go out and buy a philosopher’s hat just so I can eat it.

You have a short sighted view of what philosophy is. Good philosophers do look at the world and do care what is truly the case. They don’t see themselves in competition with science. They happily embrace what science provides. And good scientists embrace what philosophy provides. And sometimes it’s hard to draw a line between science and philosophy. Was Einstein a philosopher or a scientist?

I wouldn’t want to defend Aristotle, but I would argue Spinoza has been more important to mankind than anyone you would designate a scientist – not least because all good scientists have a metaphysics, the metaphysics of Spinoza.

And as Ant slyly recognizes, you have been making philosophical arguments, and will continue to do so if you are to make any sense at all.

Replace “philosophy” and “philosophers” with “theology” and “theologians” in what you wrote, and it reads just as well.

Which is exactly my point.

You can do philosophy with or without a rational analysis of objective observation, but science is nothing but. And, just as theologians routinely attempt to usurp the achievements of science for their own by declaring all science merely the exploration of the greater glory of the works of the favored pantheon, so, too, do philosophers take pride in the conceit that science is merely natural or applied philosophy.

Yes, yes. In ancient times science grew out of philosophy — but so, too, did astronomy grow out of astrology, and nobody would take seriously an astrologer’s claim that NASA is an astrological endeavor.

Again, we can make an objective observation as to activities that are philosophical and which are scientific, and it’s the rational analysis of objective observation where you’ll empirically find the dividing line.

Cheers,

b&

P.S. The thought that Spinoza is more important than Newton or Darwin or Curie or is…mind-bogglingly arrogant. Indeed, there’re scores of scientists more important — Ada Lovelace and Emmy Noether alone, if you just want to restrict it to those who did foundational work on theoretical frameworks. And if it’s something akin to “metaphysics” you think significant, that would be Laplace and his Daemon. b&

You ask:
“How do we know which philosophical notions are and aren’t correct? Do we further philosophize about them, or do we go out and check the notions against the evidence?”

How do you know any particular observation or evidence proves anything? Maybe it just a one off, or million off result. You need philosophy to deal with that.

I see the problem as defining what knowledge is. We can have no certain or justified knowledge at all – we have conjecture and criticism. Knowledge consists of the conjectures that stand up to criticism (until they don’t). Even in what you want to call science, before you can even know what experimnents or observations to make, you first need a conjecture.

You also say:
“If you can point to a single philosophical notion deserving of serious consideration that is not supported by equally-serious observation, I’ll go out and buy a philosopher’s hat just so I can eat it.”

This is just a bad understanding of what philosophy is. Of course philosophers will welcome observations supporting there notions. They will need philosophical arguments to explain why the observations support their conjecture. For example, to offer a “serious philosophical notion” I’ll take “God does not exist” – this has withstood heavy criticism and is still standing, so I count it as knowledge, until the unlikely event some criticism actually stands up against it.

How do you know any particular observation or evidence proves anything?

“Proof” in this context is a classic and incoherent philosophical superstition. You might as well substitute theological discourses on the meaning and purpose of life.

For example, to offer a âserious philosophical notionâ Iâll take âGod does not existâ â this has withstood heavy criticism and is still standing, so I count it as knowledge, until the unlikely event some criticism actually stands up against it.

Centuries before the Caesars, Epicurus offered up the empirical observation that there are no powerful moral agents with the best interests of humanity at heart. As I reframe it, why don’t the gods ever call 9-1-1?

As it so happens, “god,” is, itself, an incoherent and logically inconsistent term. Yet, much of non-Newtonian physics is inconsistent with classically-understood ancient philosophical logic, so the incoherence should merely be taken as a strong indication that the phenomenon under consideration doesn’t exist, at least not in the form proposed. Our overwhelming confidence of the conclusion might start with the illogic, but the slam dunk comes from the neverending mountains of evidence contrary to any and all theological claims.

…to take but one more modern example, pretty much every modern deity is a creator deity philosophized into existence through the Aristotelian superstition that abhors infinite regress. “First Cause” is the common label. And when you compare philosophical and theological superstitions about the origins of the Universe with modern cosmology and multiverse theory…

I’m a philosopher who thinks metaphysics is extremely important, and Spinoza is one of the greats. But I also believe Emmy Noether has *also* contributed to metaphysics, by also discovering some extremely general truths about reality.

The problem is that “metaphysics” is itself as incoherent a concept as the divine and primum movens. It presupposes that there is some ultimately-fundamental nature which could be known.

In stark contrast, ever since Turing and GÃ¶del, we’ve had reason to be overwhelmingly confident that such knowledge is impossible (you can never rule out a conspiracy) such that even the very concept is incoherent.

You could, after all, be a brain in a vat, and the vat could be a subroutine of the Matrix, which could be a program running on the Holodeck, which could be the subject of one of Alice’s Red King’s Dreams, which could be an hallucination induced in a wino by the CIA using Martian mind-ray technology.

What we can do is learn all we can about reality as we observe it. And, even should some conspiracy theory hold, we might even be able to gain some knowledge of it — but not if the conspiracy is “good enough.” But to go beyond that is to engage in useless theology. Your search for “metaphysics” is a search to uncover the ultimate conspiracy underlying all other conspiracies…and, by its very nature, your search can’t even hypothetically involve investigation of observation.

If you do want to gain some understanding of reality as it presents itself, including questions of origins…well, the cosmologists are way ahead of the philosophers.

We have good reason to have some, not much but some, confidence in modern multiverse theories of cosmogenesis. And we have absolutely zero reason to take seriously any proposals for ontologies more fundamental than that — at least, not until we have good reason for very strong confidence in an explanation for the horizons we’re currently probing.

“I’m a philosopher who thinks metaphysics is extremely important, and Spinoza is one of the greats. But I also believe Emmy Noether has *also* contributed to metaphysics, by also discovering some extremely general truths about reality.”

I remember Noether (very vaguely) from grad school algebra as being involved with Ring Theory. Is she the same?

I see Spinoza as the key historical figure in bringing free, secular governments into the world. He wasn’t alone in this, but I would rank him highest.

Carl: Noether proved that for every conservation law there is a corresponding symmetry in (the property of) things. I know she did work in algebra as well, so your remark about ring theory makes sense.

Yes — and the scientific principles we have the most confidence in, the ones that have never demonstrated even the slightest hint of inconsistency nor unreliability, are those about conservation. That’s a big part of the attraction of Lawrence Krauss’s “Universe from Nothing”: the total energy of the Universe is zero, so everything is conserved.

(One might wonder how energy can be zero but still lots of stuff driven by energy can happen. The answer is that stuff isn’t actually driven by energy, but by entropy…but that’s a topic for another day.)

Thanks to Noether, all of physics can succinctly, poetically, and accurately be described as a search for symmetry. Find the symmetries and how they break and you’ll understand what’s going on.

In stark contrast, from Spinoza’s metaphysics, we get nonsense such as this:

By substance I understand what is in itself and is conceived through itself, i.e., that whose concept does not require the concept of another thing, from which it must be formed.

Pure Aristotle, the special pleading of an unmoved mover to stop that evil infinite regress. We’ve no sign that there actually is anything fundamental, and there isn’t even any way in principle that you could determine whether or not whatever you think is fundamental actually is. Even now, with the completion of the Standard Model thanks to the discovery of the Higgs by CERN’s LHC, we’re highly confident that there’s still physics beyond the Standard Model to be found — meaning that the bosons and fermions of the Standard Model aren’t themselves fundamental Spinozan “supstance.”

By attribute I understand what the intellect perceives of a substance, as constituting its essence.

Pure dualism. The Universe gives not one whit what we do or don’t think of anything, and our perceptions influence reality not at all. (At least, no differently from any other physical process.)

And he ties it together thusly:

Except God, no substance can be or be conceived.

Seriously? That may well have been fine and dandy a few centuries ago, but today?

And his philosophy of mind is…incoherent. “The human Mind is a part of the infinite intellect of God” and similar such nonsense. We’re talking Chopra levels of woo.

Whatever influence he might have had on earlier scientists, there’s absolutely nothing of Spinoza in modern science in any meaningful form.

Spinoza may sound like Aristotle, but only because the vocabulary of Aristotle was what he had to work with. What Spinoza did (continuing what Descartes and others began), was break the hold Aristotle and Christianity had on Europe for preceding centuries after the church had nearly erased all trace of Epicurus. You can see it as an Epicurean revival.

The quotations you cite sound damning, but in the context of his full argument they make eminent sense (unless you are a theist). In modern terms, Spinoza sets out to say there is nothing outside the universe, nothing supernatural, and no god. Knowledge does not come from authority (church, Bible, or Aristotle), it comes from good explanations.

Spinoza’s most important contributions (in my view) are in political philosophy – his influence on modern liberty and modern government. The neuroscientist Antonio Damasio has written “Looking for Spinoza: Joy, Sorrow, and the Feeling Brain” praising the philosopher’s relevance to what he is discovering in his lab. Spinoza is widely recognized as the founder of modern Bible criticism, employing historical, literary, and linguistic techniques used by religious and secular scholars alike – even most religious scholars now must agree that the Bible is a thoroughly human book. Imagine if the Islamic world had this view about the Koran, or the West still held the pre-Spinoza view.

Most influence Spinoza has on cutting edge science, will have to be the sort he had on Einstein, which in Einstein’s own estimation was profound.

Ben, a serious question. You are obviously very smart and very well read. What have you read of or by Spinoza? How did you form your opinions about him?

You write after quoting Spinoza’s definition of “attribute:”
Pure dualism. The Universe gives not one whit what we do or don’t think of anything, and our perceptions influence reality not at all. (At least, no differently from any other physical process.

Spinoza is generally considered a monist, the exact opposite of a dualist. The rest of your words here are pure Spinoza, and not in a peripheral manner. He says exactly this, if not verbatim, clearly enough no one would find a difference.

Ben, a serious question. You are obviously very smart and very well read. What have you read of or by Spinoza? How did you form your opinions about him?

Honestly? Not much.

But I haven’t read much of Aquinas, either. I ate up Lewis’s Narnia series as a child, but haven’t read much of his straight-up apologetics. My reading of other “serious” theologians is equally slim.

And for the same reason. When the go-to money quotes in Wikipedia and similar sources are so bad, and when those quotes match the overall picture painted…what’s the point? Life’s far too short to waste on such nonsense.

And, even more to the point, what matters is not who said it, but what is said. It takes me conscious effort to remember that Eratosthenes was the one who came up with the sieve for finding prime numbers — but I used to use that algorithm as an assignment for an introductory programming class I taught many years ago. So I really don’t care that it’s Spinoza who thinks that our minds are parts of an overarching divine supermind; such superstition is long-since-debunked nonsense right up there with the Philosopher’s Stone.

I note that, in your defenses of Spinoza’s importance to science, you’ve yet to point to any concrete facts about nature which he put forth. In stark contrast, I gave you multiple examples such as Noether, who identified the connection between conservation and symmetry, which is the very essence of all modern physics. Or Laplace and his Daemon, a paradigm that has held fast ever since. (“Give me the complete current state of the system and all the rules by which it functions, and I’ll give you all past and future states.”) I could keep going — Heisenberg who identified the inherent limits of resolution, Einstein who demonstrated the lack of absolute reference frames, and so on.

If you really want to convince me of how essential Spinoza was, you could offer up some such examples…

…but I think we both know they’re not forthcoming, else they’d be as much at the tip of everybody’s tongue as Newton and Mechanics or Darwin and Evolution.

It’s not merely that I equate theologians and philosophers. It’s that apologists for both equally fail to produce evidence supportive of their claims — merely assertion and argument, but no evidence.

Claim: Spinoza “broke” the Bible. He undercut theistic religion by demonstrating, the Bible was merely human literature. I hope I don’t have to give evidence that this was important to the scientific and political revolutions that followed.

Evidence: Spinoza’s Theological-Political Treatise, using linguistic, textual, and historical evidence, has now gained wide acceptance – both his conclusions and his methods, even by theologians and Bible Scholars.

Okay, that’s well and good. But it’s religion and politics, purely human endeavors, and not even remotely hypothetically tangentially related to our understanding of reality.

It’s also far from new. Anybody throughout history who wasn’t an Abrahamist could have told anybody who asked that the Bible was merely human literature.

You’d probably argue that Jefferson was inspired by Spinoza and so Spinoza gets all the credit, but I’d argue that Jefferson and his “Wall of Separation” did far more than Spinoza to free the West from the shackles of Christianity. And let’s not forget the Jefferson Bible, which is much better known today than any of Spinoza’s works.

“maybe the true surprise is that Pythagorean triangles are expressible rationally….”

I suppose, if you’re fiddling around adding up pairs of squares, you’re likely going to come across a few cases where they add to make another square, just on the odds. On that assumption, it would be odd if there were no Pythagorean triangles.

Yes, the set of rational numbers has measure zero in the real line, so it is unsurprising that pi is irrational, and in fact (as pointed out above), transcendental.

In fact, if one were to just randomly pick out any number in the real line, the probability that the said number would be irrational is 1.

This whole question just demonstrates our prejudice toward the rationals.

If I remember the story, as Asimov to.d it, iirc, Euler was debating with an atheist, who had prepared compelling philosophical arguments for God’s non-existence, but knew no math[s], so was too flummoxed by Euler’s proof to continue.

Kind of like WLC (until he came up against Sean Carroll).

No idea, but it is cool that we know that it is, rather than not. As an experimental physicists, it’s irrelevant to me. I can define 2 as 2.00000…. with an infinite number of zeros and it is now as long as any irrational number, including pi. To me, all numbers are the same. And 1/0 = infinity (for any rigid math teachers out there).

Why are there three quarks to a proton? Why is the lifetime of 22Na about 2.6 years? These questions may have specific coherent reasons, but all the answers are ultimately tied to some arbitrary way in which nature is.

“I can define 2 as 2.00000…. with an infinite number of zeros and it is now as long as any irrational number..”

Right, for sure.

But you could even have defined 2 to be 1.999999…., in case somebody quibbled that 0’s are not fair. Or by using base 3, as 1.222222…., as well as 2.000000… of course. So your ‘unfair’ opponent has nothing left to buttress their argument.

In any case, further above several people are a bit misleading if they are implying at all convincingly, either via Cantor’s cardinals, or even via measure theory I think, that \pi is irrational because the probability is zero that a ‘randomly’ chosen number is rational.

One reason is that \pi is computable and the set of computable numbers is merely countable. (After all, you can easily ‘infinitely list’ all programs and so all which happen to run forever spitting out a digit every so often.) And I think it is again merely a set of measure zero, but haven’t checked, so that puts doubt on the measure argument as well.

Additionally, as David Wallace has cogently argued in his long Born rule section in the book on Everett’s quantum many worlds, a convincing meaning of the word “probability” is still very far from clear to either scientists or philosophers. And if anything that is easier to make progress on in the Everett multiverse.

Finally to another above, I’d argue for ‘e’ as being the more basic than \pi in the fame contest for which transcendental number is the winner. My reason is that ‘e’ arises from a 1st order DE (differential equation), namely y’=y,
whereas \pi from a 2nd order, namely y”=-y.

There are *many* more irrational numbers than rational numbers. One way to look at this is that if you look at the rational numbers in the interval from 0 to 1, they take up, mathematically speaking, zero amount of space. They’re countable, i.e., they can be put in one to one correspondence with the positive integers, while the irrational numbers, as Cantor famously proved, are uncountable.

So it would actually be surprising (and mean something) for an arbitrarily chosen number to be an integer, or even rational.

Pi (and e) are more than irrational – they’re also transcendental. That means they’re not roots of a polynomial with rational coefficients. The square root of two, again famously, is not rational. (You can prove that by a wonderful technique called infinite descent.) But it’s not transcendental: it’s a root of the equation x^2 – 2 = 0

Granted that an arbitrarily chosen number is overwhelmingly likely to be irrational. But doesn’t that just beg the question of whether pi counts as an arbitrarily chosen number?

Did you know that if you raise e to the power of pi and then subtract pi, you’ll get exactly 20? It’s a good way to check how well a computer program or spreadsheet handles high-precision arithmetic.

Possibly trivial semantic observation, but my mnemonic for recalling the meaning of irrational numbers is to remember that the opposite of irrational is rational and that rational numbers are those that can be written as the ratio (instead of saying fraction, even though they are effectively the same) of two integers. An irrational number cannot be written as the ratio of two integers. For example, 3 is a rational number because it can be represented as the ratio 18/6.

I have a somewhat different take. Pi is not just an abstract property of Euclidean geometry. It shows up in quantum mechanics.

The fact that it is transcendental suggests to me that reality is analog rather than digital. In other words, not the matrix. Reality is not countable.

Way-back-when in grad school I decided that integers were artificial constructs (and therefore so is the concept of “irrational”) and that mathematics in general doesn’t jibe well with reality because the closer mathematics came to describing reality the more complicated it got, e.g. Newtonian vs Relativistic physics.

Like a smarter guy said: “As far as the laws of mathematics refer to reality, they are not certain, and as far as they are certain, they do not refer to reality.”

BTW – one on the most amazing, yet simple equations in mathematics is Euler’s equation:

e^(pi*i) = -1

(Looks cooler with superscripts and math symbols!)

That equation puts together two transcendental numbers and one imaginary number, and gives you back an integer.

Geometrically, it says that if you take 1 (thinking of it as lying on the x-axis in the plane) and rotate it 180 degrees around the origin (Pi radians), you end up at -1.

BTW, the fact the Pi is the ratio of the circumference to the diameter of a circle is lurking around in the back of that proof, though it takes a bit of looking to find it. I has to do with how the function sin(x) is defined and the units used for the function (radians).

The answer to Jerry’s question is that there is a difference between proving that π (the mathematical constant) is irrational (transcendental) and explaining _why_ the ratio of the circumference of “standard” circle to its diameter is that number. Just as he wrote, a geometric step is required to link these.

Someone already said that in a curved space the ratio is not π. A nice example is the sphere. If you are limited to two dimensions and measure distance along the surface, then the diameter of the equator consists of two half of meridians. The ratio C/D is then exactly 2.

What we commonly mean by the circle is a curve equidistant from a point on a plane with Euclidean metric (which roughly means flat space). It then so happens that the position of all points on the circle can be given by (x,y) = (r cos(t), r sin(t)), r being the radius and t going from 0 to 2π – precisely the mathematical constant, which must be the period of the trigonometric functions. (Note this is not the only possible parametrisation.)

The integral formula for the length of such curve uses the derivatives of the above and is C = ∫sqrt(r^2 sin^2 t+ r^2 cos^2 t)dt = 2πr, thanks to the known trigonometric identity.

I would have to sit down and try to write it but if pi were rational, then a circle would have to be a polygon, wouldn’t it?

For example, if pi were 3, then a circle and the inscribed hexagon would have the same perimeter, which would mean they are identical.

I am fairly sure for any rational number p < pi there is an inscribed polygon that has perimeter r*2*p … since all inscribed polygons perimeter is supposed to be smaller than the circle's …

I think that is it exactly. If 22/7 were right, you could have a 22-agon shape with a “diameter” of 7. The same would be true of any other ratio. Therefore pi must be irrational.

The conjecture is true if you allow irregular polygons (and insist p > 0). A proof could go like this. Pick p r*2*p. Fix one vertex and slide all the other vertices of the n-gon along the circle toward the fixed vertex. The perimeter of the n-gon with slid vertices will shrink continuously toward 0 (picture all n vertices almost coinciding). The continuous shrinkage from q to 0 insures at some point the perimeter will equal r*2*p, since q > r*2*p > 0.

“for any rational number p 0). A proof could go like this. Pick p r*2*p. Fix one vertex and slide all the other vertices of the n-gon along the circle toward the fixed vertex. The perimeter of the n-gon with slid vertices will shrink continuously toward 0 (picture all n vertices almost coinciding). The continuous shrinkage from q to 0 insures at some point the perimeter will equal r*2*p, since q > r*2*p > 0.

for any rational number p 0). A proof could go like this. Pick p r*2*p. Fix one vertex and slide all the other vertices of the n-gon along the circle toward the fixed vertex. The perimeter of the n-gon with slid vertices will shrink continuously toward 0 (picture all n vertices almost coinciding). The continuous shrinkage from q to 0 insures at some point the perimeter will equal r*2*p, since q > r*2*p > 0.

For some reason, what I’m trying to post keeps getting the middle cut out. I’ll try splitting it in two.

Part 1:

Great to see so much math interest here.

The conjecture

“for any rational number p < pi there is an inscribed polygon that has perimeter r*2*p … since all inscribed polygons perimeter is supposed to be smaller than the circle's"

is false if you stick to regular (all sides equal length) polygons:

An n-sided polygon will have some perimeter P1 strictly smaller than P2 the perimeter of an n+1 sided polygon. The infinite number of rationals between P1 and P2 will never be the the perimeter of a regular polygon.

Well, it works if you allow for open inscribed polygons, right?

So, for any fraction p/q < 2pi, you can build a polygon (open or closed) where all vertices lie on the circle of radius 1, and the length of all the sides is exactly p/q.

Since it won't be a circle, because distance to the center is not constant, it shows no p/q < 2pi works, therefore pi can't be expressed as p/q?

I haven’t watched the video yet, but here’s my guess.

Pi is just a special case of calculating the length of a curve given by some function f(x). We can do this by taking the first derivative of f, applying the Pythagorean theorem, and integrating sqrt(f'(x)^2 + dx^2). In effect we’re adding up an infinite sum of infinitesimal hypotenuses.

Since this involves taking square roots, we should expect the result to be irrational, except in special cases constructed around Pythagorean triplets.

Any real mathematicians out there should feel free to poke holes in this argument.

“And I’m a bit surprised that nobody uses the irrationality of π as an inexplicable fact about mathematics that implies the existence of God.”

In the book version of the 1997 scifi movie Contact, (not in the movie) something like that was at the end: It was discovered way… way… way out in the decimal part of pi the numbers started a sequence or code that would plot up as geometric shapes (or something like that). It was implied that it was the “signature” of the creator, and purposely built into the geometry of the universe: not necessarily god, but an advanced civilization that was creating universes through big bangs.

It was implied that it was the “signature” of the creator, and purposely built into the geometry of the universe:

From the second link above:
++
The problem is that pi is not a physical constant, it is a mathematical constant, the ratio of the circumeference of a Euclidean circle to its diameter. It is not adjustable because its definition makes no reference to reality. Furthermore, there is no distinct physical constant that would be meaningful in any universe even vaguely resembling ours. In curved spaces, there is no constant ratio of circumferences to their corresponding diameters. The small circle limit, where no singularity is involved, is precisely pi, hence the qualification “distinct”.
++

Useless toy #437 (using the same numbering scheme as for useless facts), the first occurrence of my (UK, mobile) phone number in π is at position 52216852 after the decimal point.
You can go one way, but not the other, with http://www.angio.net/pi/ and probably other tools. Unfortunately, it won’t quite stretch the 3 extra digits necessary to give the number fully, because it only looks at 200 million digits of π .

There are some very sharp people commenting here so I will ask a question that bothers me occasionally. I think I read somewhere that (A) the decimal digits of pi are proven to be a random sequence which (B) implies any arbitrary sequence of digits (however long) *will* appear (eventually) in pi (as expressed in decimal digits).

I would love to know whether either (A) or (B) are true, and especially whether (A) and (B) is true.

If (A) and (B) are true, I suspect this is one of those things in which theory and practice don’t intersect because of the practical difficulty you’d need untold googols of universe lifetimes to find the billion digit sequence (which encodes a high def picture of Hili the cat say) within the digits of pi.

(A) It depends on what you mean by “random”. In fact, I would say that random is probably not the best word in any context. It might be more appropriate to say the digits are “arbitrary”.

(B) is something *much* stronger. It is certainly not true that (A) implies (B) in general; not for pi, not for sequences generated by sampling digits 0-9 uniformly at random. It is strongly suspected that pi has this property, but it is not known.

A) >It depends on what you mean by “random”.
Indeed, it is difficult to express exactly what I mean because clearly the digits are absolutely not random – they’re the digits of pi! I wish I had my original source reference – I would speculate that it would be something like “the digit stream specifies to arbitrary accuracy the mathematical constant pi but in all other respects has every known property of a stream of random digits”.

B) I am afraid I don’t follow you here. Are you saying that an infinite stream of random digits does not necessarily include any arbitrary finite stream of digits? That would be disappointing. Remember that infinity is a lot :- ) Also when you say it is strongly suspected that pi has this property, but it is not known, which property are you referring to?

There’s another property, called “normality”, IIRC, which means that any finite sequence of digits also exists. I do not know whether it has been proved that pi is normal.

(Why are we doing this all now? The next science holiday is mole day, not pi day!!)

Thank you, “normality” turns out to be the key concept here. Some googling turns up the information that pi is not proven to be normal, although it is strongly conjectured to be so. The money quote (in Wikipedia) is ‘In particular, the popular claim “every string of numbers eventually occurs in π” is not known to be true.’ As far as I can tell this is equivalent to restating that pi is not (known to be) normal. Basically normal means the digits (in any base) are randomly distributed. It *is* known that almost all real numbers are normal, so almost all real numbers do have a HD picture (video even) of Hili the cat in there somewhere, if you just go far enough.

Damn, my head just exploded and now someone (else) is going to have mop all the gore off the screen and desk. Or are they. Can that really be true? I know there are different types of infinity, some stronger than others – but these are the same types of infinity. Clearly finite can be a subset of infinite. And I can believe weak infinite can be a subset of infinite. But how can peer infinite be a subset of infinite? I’m not even going to try and think about the recursive aspect of your comment. As Clint Eastwood would say “A man’s got to recognise his limitations”.

There was a project some years ago which used some fancy maths to calculate the (IIRC) quadrillionth (binary) digit of π without calculating the previous [lots of] digits : pihex. It was a distributed computing project that used 1900-ish computers from a little shy of 700 workers (of which I was one, contributing two computers when I was onshore). In particular, it was carefully designed to not need internet access unless a new work unit was requested manually, which was just as well since it was before I had regular internet access.
Given that such weird maths is possible, I wouldn’t even touch a bet that it was impossible to find an arbitrarily long (binary) string in π , though the calculation may take some time and require substantial intermediate storage.

Dare I mention Mandelbrots? A graphical representation of the weirdness produced by digging deep into an algorithm.

What fascinates me is that you can take a good computer program and zoom in and in to an arbitrary point and what you see has probably never been seen by anybody else ever and may never be. Even though anybody else navigating to that exact same point under the same conditions would see precisely the same thing. But there is so much complex detail, I think the likelihood of anyone else ending up at that precise location is vanishingly small.

(I suppose it’s a bit like picking up a grain of sand on a beach. The odds of anyone else picking up that exact grain of sand and examining it are minuscule).

Having seen a few beaches excavated and exported to Saudi Arabia, I think the odds of the sand grain having been handled before are better than those for finding a previously-viewed grain of the Mandlebrot set.
I did program a Mandlebrot viewer some years ago. Even without having access to double-precision numbers (this was BASIC, after all), I was unable to find limits to the depth I could get to.

I was a little bit careful to say ‘picking it up and examining it’. I originally said ‘seeing’ but, standing on top of a cliff, I could technically ‘see’ every grain on the surface of the beach.
But I do agree with you, I think the Mandelbrot set is (probably) potentially infinite in its recursion. I’m not so sure about actual Mandelbrot computer programs, I think it eventually reaches a limit. In ‘Xaos’ for example, I seem to have reached a limit at a radius (view diameter) of 5e-14.

P.S. I did write a program for Mandelbrots, in BBC Basic 5 for the Archimedes. That language was almost uniquely well-suited to Mandelbrots, since it had a built-in assembler that could be called from within the program. So one could write the routines for selecting the screen area, display etc in easy-to-program Basic, and the short but multiply-recursing (many hundreds of recursions per point) calculation loop in much faster assembler.

Pi has lost its shimmer for me over the years, jaded by grad school perhaps. What fascinates me though is that irrational numbers are so damn hard to find in practice, yet they are everywhere in theory. As several commenters mentioned above, there are far more irrationals than rationals (in terms of set cardinality); you can say that there are as many irrationals as real numbers, but only as many rationals as integers. Yet it is notoriously difficult to find most of those irrationals.

Sure, we all know about the various algebraic irrationals (various roots of integers), but this is an extremely small set – also only as big as the set of integers. Where are all those other irrationals? We see pi and the Euler constant e all the time. And you’ll see some other less common ones like phi for the golden ratio. But that’s about it. Pi pops up everywhere because of its intimacy with the circle, and e pops up everywhere because of its closeness with the exponential (limit of (1+1/n)^n for example). But write down a closed form expression for me that defines an irrational number that isn’t just a rational scaling of one of the common ones. Extremely difficult to do.

Also, the irrationals aren’t closed in any coherent way, which can lead to all sorts of funky things like the fact that there exist irrationals a and b such that a^b is actually rational. Or what about the fact that we don’t even know if pi + e is irrational?

It’s no surprise to me that many mathematicians are deeply religious. I met many of them as a grad student. It’s a beautiful subject, but its beauty can be absolutely blinding.

At least with respect to outside of USians, I very seriously doubt that. And though it might be hard to get the evidence, I’d be willing to bet that, within the U.S. National Academy of Sciences, the mathematicians are no more religious than the rest, religiosity being quite rare there.

There seems to me to be a kind of extreme dislike of anything slightly like Platonism, pretty shallow I’d say in many cases. Then the logical mistake is made that ‘belief in existence of something outside the natural world implies belief in some kind of god’, whereas it’s the converse of that which is pretty obvious. That quoted belief, which is nonsense, then combined with the fact that many mathematicians adopt a Platonist philosophy of mathematics, leads to acceptance of the quote at the top.

It is interesting here that many refer quite straightforwardly to irrationals, to cardinality, to measure theory. I wonder how many, if not platonists, have thought much about what they mean by those words (except Ben Goren, who I’m quite sure has thought seriously about that.)

That statement was meant to be taken relatively; i.e. many mathematicians are religious relative to the proportions who are religious in the other sciences.

My evidence is largely anecdotal, although I do recall seeing several surveys that polled religious belief among different scientific fields. Mathematicians were always up in the highest category of religious belief. I do not recall sources though, so you can either try to Google them or not believe me.

I don’t appreciate the implication that I don’t know what I’m talking about. You could very easily click on my name and go to my website, which would corroborate the fact that I’ve spent plenty of time thinking about measure theory and the like.

No offense, just asking for evidence. If Googling for surveys will produce such evidence, perhaps it’s the asserter of that claim, viz.

“It’s no surprise to me that many mathematicians are deeply religious. “,

who would be kind enough to find it, not the person (me) who has expressed skepticism.

I will say however that my last two paragraphs previous didn’t, and were not intended to, refer specifically to you. I have no idea what position you take, platonish or not, with respect to a philosophy of mathematics.

“But if there’s some proof out there that the ratio of a circle’s circumference to its diameter, based on the geometry alone, must be irrational, I’d like to know about it.”

Not a proof, but I have thought about this so I will share my layman reasoning.

If you describe a circle as a polygon with infinite sides then it kinda makes since that pi is an irrational number when the number of sides cannot even be expressed as a number.

I was thinking on those lines myself.
Once you get above a certain integer “n“, then an n-gon can be inscribed within a circle, while an (n+p)-gon (p is also an integer, and might be 1 or higher ; if it’s 1, you get as tight a fitting of inscribed-gon, circle and escribed-gon as is possible ; if it’s 2, then you don’t have to worry about even integers in your ratio)would be escribed by the same circle (call these the in-gon and e-gon).
If you sketch the relations between the in-gon, the circle and the e-gon, you can see that by adding 1 side to the in-gon and 1 (or more) side to the e-gon, you’d get a better fit between the three figures. (I’m pretty sure I’m stealing this from Archimedes’ method for calculating a value for π so I’m feeling pretty good on sketching the proof this far.)
Our in-gon and e-gon represent the numerator and denominator in a rational approximation to π
By adding to the in-gon and the e-gon, we get a better approximation to π involving larger numbers.
I can’t see any way that you can stop this process of improving the fit, for any n, so I can’t see that any ratio of integers can produce a rational approximation to π which cannot be improved by this process. And I think that’s a mathematical hop, skip and a jump from “therefore π is irrational, QED.”
I could probably tighten that up a lot before I run out of maths, but I’d need to learn HTML maths mark-up far better than I know it now to express that, and I’d probably break WordPress’ rendering engine in the process.

Yes, the ratio of the perimeter to the long diagonal of a regular n-sided polygon approaches pi as n increases. Incrementing n increase the precision so the digits cannot terminate and the sides and diameter of the polygon change so the digits cannot end with an infinitely repeated sequence (waves hands.) Ergo, irrational.

Yes, the ratio of the perimeter to the long diagonal of a regular n-sided polygon approaches pi as n increases.

All diagonals, not just the longest (nor the shortest) approach π.

That’s another way of looking at it. Were π rational, that would imply that all arbitrary diagonals of all polygons are also rational, which would further imply that all numbers are rational.

I’m missing the argument here. A sequence containing irrationals can have a limit which is rational; for example, sequence with nth term being 1/(square root of nth prime) has the rational number 0 as limit.

So you seemed to me to be saying a rational limit for a sequence implies terms of the sequence are also rational. But it’s late, and maybe I’m just missing the point.

“Pi is irrational” is a mathematical truth. Like all provable mathematical truths or theorems, you make definitions and axioms then by deduction prove the theorem. All provable mathematical truths are of the same nature as “there are 36 inches in a yard” – they are in a very real sense just tautologies. Some are just more complicated to see than others.

I’m using “provable” restriction above, because Godel has shown there infinitely many unprovable truths in any sufficiently complex mathematical system.

“And then the astute theomathematician could bring up the square root of two. . .”

I’m late to this, but still surprised no one ran with this a bit more. There’s the root of -1.

I’m convinced there is a really good joke in the window/bumper stickers I’ve seen lately of “HE>i” from the clothing company hegreaterthani.com – them saying its based from John 3:30. “He must increase, but I must decrease.”

Wouldn’t it be more accurate simply to say:
HE = i
i = -1^0.5
HE = imaginary

If I were just to throw up my own sticker with “HE=i” people would just think I was boasting and wouldn’t get the joke. That’s why I think it’d be better as a proof. But still one more subtle than the above that gets the observer to conclude the imaginary part.

There is something funny there. I just can’t write it right.

If you think I’m claiming Spinoza is *exclusively* responsible for so many good things, I am not. Spinoza is part of a radical tradition, including (not exhaustively) Epicurus, Lucretius, Machiavelli, Bruno, Descartes, Hobbes, Locke, Hume, Franklin, Paine, and Jefferson – who all played important roles in creating the modern world. As did Galileo, Newton, and many others.

The foremost thing done by these radicals was to discredit the idea there is some authority in regards to knowledge, which pre-Enlightenment was Aristotle and the Bible. They broke this stranglehold. The basic idea that we live in a natural, explainable world emerged – where science and evidence was more and more valued.

You write:

“It’s also far from new. Anybody throughout history who wasn’t an Abrahamist could have told anybody who asked that the Bible was merely human literature.”

If it hasn’t been clear from context, I’m talking about the West. Spinoza was *the first* to publish this claim, and as seems to be important to you, give copious evidence.

As to Jefferson, not only did his Library contain the works of Spinoza, he lived in an intellectual environment saturated by Spinoza. Spinoza, of course, doesn’t “get all the credit” but if you know the history he deserves a great deal, as do many others: Bolingbroke, Sidney, Chubb, Shaftsberry, Toland, Locke. Almost all largely forgotten, except Locke. You have admitted you know little about Spinoza, so why do you continue to make statements about him? I doubt if you are interested, but others might find this list of books useful:

By the man himself:

Ethics (1677): (avoid the Elwes translation, look for something done later like Curley or Shirley): This book is notoriously difficult and hardly anyone will jump into it and comprehend much.

Steven Nadler (2011):
A Book Forged in Hell, Spinoza’s Scandalous Treatise and the Birth of the Secular Age.
A very readable, entertaining look at the TTP, the circumstances and immediate aftermath of its writing.

Rebecca Goldstein (2009):
Betraying Spinoza, The Renegade Jew Who Gave Us Modernity. Very readable, gives an overall picture of Spinoza (by a philosopher who had a Ben-like attitude until about the third time she had to teach a Spinoza course), along with some speculation by a top novelist.

Matthew Stewart (2014):
Natures God, The Heretical Origins of the American Republic.
My favorite book. It gives a rigorous argument that America is not a “Christian Nation.” Well researched and documented – one third of the book consists of end notes. It focuses on the Declaration of Independence and the “radical” philosophers I’ve mentioned.

Jonathan Israel:
Radical Enlightenment (2001)
Enlightenment Contested (2006)
Democratic Enlightenment (2011)
You have to be pretty serious to tread here. Three thousand-page books by a historian digging through evidence from the previously unknown or obscure through to common knowledge.

The foremost thing done by these radicals was to discredit the idea there is some authority in regards to knowledge, which pre-Enlightenment was Aristotle and the Bible. They broke this stranglehold. The basic idea that we live in a natural, explainable world emerged â where science and evidence was more and more valued.

Just to put things in perspective, you started this diversion by claiming that Spinoza was a more significant figure than any scientist. Now you’re including him as a minor standout of many peers in a long laundry list of antiestablishment figures. And your evidence largely consists of a list of works that you yourself describe as “notoriously difficult” and borderline incomprehensible.

You’ll excuse me if I’m decidedly underwhelmed by this retraction, and also if I’m struck by the obvious parallels with Christian apologists insisting I haven’t correctly read the right theologian to properly appreciate the true sophistication of their arguments.

I made no retraction. I stand by the claim that Spinoza was the most important figure. You transmogrified my claim to something like “the only important figure” and I was pointing out my claim was not so absurd. And you misunderstand me if you think I’m saying he was a “minor standout of many peers” – I think he was the central figure in that list of greats, the one who crystallized what came before and deeply influenced and set the tone for those who came after, along with much of western culture.

That’s my view, Neil deGrasse Tyson thinks it was Newton, and many probably would pick Darwin. Good choices, but I disagree. Maybe I could be convinced to change my mind, but this opinion has been jelling for 50 years, so it’s unlikely.

I’m not so much arguing you haven’t read the right thing, which is not in itself an invalid argument, I’m pointing out what you have already admitted, you have read almost nothing (about Spinoza). It’s not the sophistication of any argument holding you back, it’s the utter lack of knowledge concerning the topic (Spinoza, his achievements, his views, and his influence). It’s well beyond my powers to convince anyone, using a few hundred words, of such an admittedly bold claim, particularly someone with only misconceptions about the subject to begin with. Hence the book list.

There is one book on my list I would class notoriously difficult – Ethics. The Israel trilogy is extremely long and detailed, and no doubt most will find it boring. The other four sparkle. I could add many additional titles.

I stand by the claim that Spinoza was the most important figure.

You’re certainly entitled to your opinion, but the facts are that his physics are entirely unrelated to reality and his theology and ethics substantively no different from positions expressed at least two millennia earlier by Epicurus and his followers. As Stalin would have noted, he had no divisions. As you yourself have noted, his most important work is obscure and nigh on incomprehensible.

So, at most, though he had no intellectual innovations to his name, he revived and revised some significant ancient ideas and got a lot of other stuff very, very worng whilst expressed himself badly.

In stark contrast, Newton invented (independently at the same time as Leibniz) the most-practically-applicable (still to this day!) field of mathematics, and used that invention to almost completely describe the physics that’s most overwhelmingly relevant to human scales. Darwin showed how complexly organized life can (and does) spontaneously evolve from simple disorder — thereby slamming shut the last remaining door for the gods. Louis Pasteur is directly responsible for pasteurization and vaccination, which has saved literally billions of lives, including yours. Edward Teller rewrote the geopolitical map with the fusion bomb. Kennedy put Armstrong on the moon, and Kennedy and Khrushchev brought civilization to and back from the brink of self-immolation. Stalin forged the Soviet Union. Genghis Khan grandfathered innumerable great-great-grandchildren at the same time as he initially wrote the first geopolitical map. Caesar crossed the Rubicon. Paul popularized (to a rather limited extent) an obscure ancient Jewish demigod who most recently had been no more than a footnote in Philo’s theology, thereby setting forth the roots of Christianity — and the author of Mark gave Jesus his biography.

If, amidst all those facts, you still think that a revivalist who had no clue about how reality functions was the most important figure in history…well, that’s an opinion you’re certainly welcome to. But it says far more about what’s important to you personally than to any sort of influence the man actually had on history.

You do a lot better better for yourself by proposing alternatives. However, what you write about Spinoza is deeply, factually wrong, and only displays your lack of any real knowledge on the subject. You are satisfied in your ignorance, and can’t be induced to cure it. Your loss, but it reflects badly on you.

I, on the other hand, have profound appreciation for Newton and Darwin, but rank Spinoza higher.

About all those digits – and an unproved conjecture, in case anyone is looking for something to do:

“The digits of π have no apparent pattern and have passed tests for statistical randomness, including tests for normality; […] The conjecture that π is normal has not been proven or disproven.[21]”

… about the Bible thing – I’m not going to make a big deal about how the Bible got pi wrong – some reading suggests everyone was close but not accurate for a long time. But if a bible thumper makes a big deal about how they were close, then we can say join the club – everyone was close.

[…] of Chicago) has a website called “Why Evolution is True”. He wrote an article titled “why is pi irrational” and seemed to be under the impression that being “irrational” was somehow […]

## 141 Comments

Almost all numbers are irrational so no surprises this one is.

Infact pi is also trancsendental which is also unsurprising as almost all irrationals are

“Almost all numbers are irrational so no surprises this one is.”

This is not actually true. There are infinitely many rational numbers, and infinitely many irrationals.

“Infact pi is also trancsendental which is also unsurprising as almost all irrationals are”

Again, not actually true. All transcendental numbers are irrational, but there are infinitely many irrational numbers that are not transcendental, such as the square root of 2.

There are countably many rationals but uncountably many irrationals (Google Cantor). In fact the cardinality of the irrationals is equal to the cardinality of the power set of the rationals. (If there’s an infinity with intermediate cardinality we’ll leave as an exercise for the reader…)

Sorry, but in terms of set theory, what you have said here is not correct. The cardinality of the irrational numbers is strictly larger than the cardinality of the rational numbers.

If you want a proof of this, cf. Cantor’s Diagonal Proof. Well, sort of…the proof deals with the cardinality of the real numbers and the integers. However, a bit more work will show that the rational numbers and the integers have the same cardinality, and that the irrational numbers and the real numbers have the same cardinality.

Dang…I see yossarian beat me too it again.

There are countably infinite rational (and natural) numbers but uncountably infinite irrational numbers.

True. In terms of measurement theory (part of real analysis) the set of rational numbers within the set of real numbers has measure zero. In other words, the size of the set of real numbers and the size of the set of irrational numbers is exactly the same.

In fact, Pi goes beyond just being irrational. It is transcendental, that is, it is not the solution for any polynomial P(x) = 0, where the coefficients of P(x) are integers. Note that some irrational numbers, such are sqroot 2, are not transcendental (aka algebraic).

This fact is why the classic problem “squaring the circle” has no solution.

The ever famous e is also a transcendental number.

Oh…just noticed yossarian’s post. He mentions transcendental as well.

Pi believes in god????

I believe in Pie!

Isn’t theomathematician the guy who used to be general manager of the Boston Red Sox and now the Chicago Cubs?

The field of theomathematics concerns itself with the square root of -1 and other imaginary numbers.

In addition to being irrational, pi is also “transcendental” meaning it is not the zero of any polynomial with rational coefficients. This is Not the case with square roots.

Proofs of God along these lined might be pi in the sky.

And even if pi’s irrationality were proof of god’s existence, the mathematician making the claim would still have all his work in front of him.

As Yossarian says above, pi is not only irrational but transcendental, meaning it is not the root of any polynomial with integer coefficients. Non-transcendental numbers are called “algebraic”. Sqrt(2) is an example of algebraic, irrational number.

The set of algebraic numbers is countable, meaning it has the same cardinality as the set of integers. You need to include the transcendental numbers to make the set of real numbers uncountable. This means that the vast majority of real numbers are transcendental. In fact, if you could somehow pick a real number at random your chances of getting a non-transcendental number would be zero.

Th eonly slight hiccup in the argument you and I are making is that pi is computable (a countable set) unlike nearly all irrationals

“eonly” – Is that some kind of Freudian slip, “e” rivaling pi as the most important transcendental, computable number?

Can the chances even be computed??

The “chances” can be proven mathematically to be zero. To do it, you first have to define things. You enter the field of measure theory. For example, all the numbers between 0 and 1 have measure = 1, all the numbers between 1/4 and 1/2 have measure = 1/4. Defining the measure precisely requires a fair bit of formal math, the examples given are just what falls out of the formal definition. It can then be rigorously proved that the measure (hence probability) of a number being non-transcendental is 0.

Interesting question… don’t know…

A rough Google search suggests 39 digits – so to 38 decimal places – is enough to calculate the … volume? … of the known universe to within one hydrogen atom.

But that leaves lots of room for the subatomic particles I guess…

Yossarian – I never thought about that. Is there a number theory picture for that, akin to how primes are the building blocks for all other whole numbers? Is it a mistake to expect geometry to have some perfect relationship with numbers?

For more pi trivia check Neil DeGrasse Tyson’s pi day tw33ts.

You only need (roughly) as many decimal digits as there are orders of magnitude in your margin of error.

If you’re using a meterstick that’s one meter long and has centimeter markings, 3.1 is about as accurate a measurement as you’re going to make. If it’s marked in millimeters, 3.14 will do the trick.

If you remember your “Powers of 10” short film from the ’60s or so, you’ll recall that there’s only a couple dozen orders of magnitude in either direction between us and the extremes of bigness and smallness. So, yes, a few dozen digits of π is ample for all engineering at any scale, and an hundred digits is beyond meaningless for even the most speculative of theoretical physics.

(Though there might be a footnote for certain multiverse theories, including the simplest which gives universal repetititition on the scale of a googolplex lightyears or so.)

Cheers,

b&

The film actually has its own website, but for whatever reason my computer is not connecting to it now.

The film can be watched Text, however.

Of course, the “Text” above should have read “here”. Sigh…

What really intrigues me about pi is that it can be calculated in several different ways that have nothing to do with geometry, let alone circles. That implies that it’s a number that has nothing to do with the properties of our universe. If there are parallel universes that have different laws of nature than ours, pi will still be the same number there that it is here.

Remember, this is coming from the same guy who thought that bats were birds!

Pi is so easily described using any number of very elegant and simple infinite series. Are these really weirder and more supernatural than 1/3 which is an infinite string of threes in base ten? http://www.geom.uiuc.edu/~huberty/math5337/groupe/expresspi.html?

Here is an interesting corollary. Imagine a circle, and imagine a string that is the length of the circle’s diameter. Place one end (let’s call it the left end) of the string somewhere on the circle, align the string to the circle, and note the location where the other end (let’s call it the right end) of the string is on the circle. Now move the string along the circle so that the left strand is now where the right strand used to be, and note the new position of the right string. Repeat the process as many times as you want, you will NEVER EVER end up on the same location. You can repeat the process an infinite number of time and each time the location at the end of the string on the circles will be different from any other locations found before. Cool way to illustrate infinity😉

That is cool!

If we use Pi = measured C/D, then in a curved space-time, Pi can be smaller. Take the sun for example: for a fixed orbital circumference, the radius will be larger than Newton would predict because the sun’s mass warps spacetime. Thus if you calculated Pi based on C/D of the sun, and you were really accurate, you would get number slightly smaller than its standard mathematical value.

But this has little to do with pi’s utility in helping to solve various problems. The easiest way to respond to the above point is to simply say ‘we don’t define pi based on an empirical measure, we define pi based on a mathematical relation…which in this case includes a flat geometry’

In curved geometries, the ratio of C/D is variable, but (barring singularities) converges on pi as D approaches zero. There is no geometry, curved or flat, in which the ratio of C/D is a constant other than pi.

There many hidden assumptions there, though, like a manifold structure etc. On a hexagonal lattice C/D converges to 3 as we reach the smallest size, and π as the size increases.

Still it’s an interesting fact for smooth cases – could you provide some references for a proof, please?

Sorry; I should have said “As far as I know there are no geometries…”

I don’t have a proof.

It might help to go back to the very basics of the meanings of the terms.

“Rational” simply means that there is an whole-number fraction, or ratio, that represents that value. You can cut one apple into three pieces; two of those pieces represent two thirds of an apple.

Were π rational, you could take a piece of string and fold in exactly so many fractions to mark off the perimeter of a circle, and fold it in some other number of fractions to mark off the diameter.

But it’s instantly obvious that you can’t do that for any small whole-number ratio, and it’s practically trivial to create other sorts of geometric objects which also don’t have any whole-number ratio relationships.

So the real question is why you should expect to be able to do this with circles.

Perhaps the better question then would be why you might expect any particular property to be expressible as whole-number ratio relationships. And, in that light, maybe the

truesurprise is that Pythagorean trianglesareexpressible rationally….Cheers,

b&

But it’s instantly obvious that you can’t do that for any small whole-number ratio,To play devil’s advocate, I don’t think so. Let’s say I measured and cut a string corresponding to C, and then marked it with evenly spaced marks. Then I use that string to measure the diameter and find out that the point were the loose end of the string crosses the circumference doesn’t align with any mark on my string. My first thought (assuming I don’t know geometry) would *not* be “huh, I guess that means no mark will ever align, no matter how closely I space them together.” Instead, my thought would be “I need more finely divided marks to get the exact measure.” I’m not sure the answer that it’s irrational is instantly obvious, IOW. Its true, yes, but not instantly obvious based on human experience or perception. Which is why the bible probably gets it wrong; because its not instantly obvious.

But what reason do you have in the first place to expect that there should be such a relationship? And how far must you go on such a quixotic quest to question your premise?

Archimedes knew that 22/7 was close but not it, at which point you should suspect that there’s no such ratio. Ptolemy pushed it to 377/120, which is more than enough to cast overwhelming doubt. By the Middle Ages there were infinite series calculation methods, at which point it should become intuitively obvious that the ratio isn’t even remotely simple.

Cheers,

b&

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But what reason do you have in the first place to expect that there should be such a relationship?I think in terms of human intuition, the idea that no matter how closely you drew the marks they would

neverline up with the circumference would be baffling and surprising. The intuitional expectation is thatsomemark will,you just have to find it.But that’s just me, Ben. Maybe your own intuition naturally leads you to the right conclusion. Even if that were the case, though, I would be wary of proclaiming your own intuited result as “instantly obvious” to people in general. I think its a very common failing for people who understand something (especially some deductive or mathematical relation) to think that what they understand must be obvious to others. It isn’t. My own personal favorite was professor at Jerry’s Alma Mater, W&M, who declared to his students (including a friend of mine) that the QM wavefunction for a particle in a box was “intuitively obvious.” Do you think he was right, or do you think he possibly made a mistake in thinking something he knew like the back of his hand must be equally obvious to others?

If deductive, logical, and mathematical relationships were instantly obvious, we wouldn’t need math classes.

“Trivial” was the favorite word to one of my grad school classmates. Every proof he new how to do was trivial – except for once in a while when you asked him to demonstrate and he’d forgotten how.

“The answer, when found, will be obvious.”

cr

I was more aiming towards the fact that we shouldn’t have

anyinitial expectations in such cases — either for or against rationality. And, once you start investigating the matter, the evidence quickly piles upagainstrationality. The ancients already knew enough to be highly skeptical of suggestions of rationality, and it was really only superstition (aka“philosophy”) that caused them to cling to rationality long after reasonably supportable.Cheers,

b&

>

“superstition (aka philosophy)” – Are you equating philosophy and superstition? Please explain.

You’ll find many people decidedly unimpressed with philosophy, especially its central conceit that one can think one’s way to understanding reality.

It is perfectly possible to be an highly-respected philosopher and to advocate positions contradicted by observation, let alone ones not supported by observation. And it is depressingly common for philosophers to disdain the notion that the obvious way to answer many questions is to go out and make observations.

Those trolley car “thought” “experiments” are the perfect example. No industrial safety engineer would even pretend to find anything useful about any of that nonsense. Psychologists learned the relevant lessons with Milgram

et al.and you’d have a damned hard time getting approval for such past an ethics review board. (Authority figure says to do something horrific and people do — we know that.) But philosophers just eat up that bullshit.So long as you care more about how you think the world

shouldbe rather than go out and have a look at what it actuallyis,you’re doing philosophy. And it should be unsurprising that the people wearing philosopher’s hats who’re doing real, meaningful work are making or analyzing observations…which is what the rest of us know as……wait for it…

…science.

And it’s been that way since antiquity. Archimedes exclaimed, “Eureka!”

afterobserving the relationship between density, weight, and volume. But Aristotle never bothered to check if things might not keep going if there’s nothing holding them back (what, not even any rumors of frozen-over lakes?), and so we got stuck with that whole catastrophe of metaphysics thatstillplagues us to this day.I mean…damn, but that’s William Lane Craig’s whole problem in his debate with Sean Carrol. WLC couldn’t escape Aristotle’s trap if his life depended on it, when the rest of the world moved on literally centuries ago once Newton got bopped on the head by an apple.

Erm…sorry for the rant….

Cheers,

b&

>

Because it’s similarly unconstrained by facts. He contends. Philosophically.

/@

No; it’s an observed conclusion.

How do we know which philosophical notions are and aren’t correct? Do we further philosophize about them, or do we go out and check the notions against the evidence?

If you can point to a

singlephilosophical notion deserving of serious consideration that isnotsupported by equally-serious observation, I’ll go out and buy a philosopher’s hat just so I can eat it.Cheers,

b&

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Ben Goren,

You have a short sighted view of what philosophy is. Good philosophers do look at the world and do care what is truly the case. They don’t see themselves in competition with science. They happily embrace what science provides. And good scientists embrace what philosophy provides. And sometimes it’s hard to draw a line between science and philosophy. Was Einstein a philosopher or a scientist?

I wouldn’t want to defend Aristotle, but I would argue Spinoza has been more important to mankind than anyone you would designate a scientist – not least because all good scientists have a metaphysics, the metaphysics of Spinoza.

And as Ant slyly recognizes, you have been making philosophical arguments, and will continue to do so if you are to make any sense at all.

Replace “philosophy” and “philosophers” with “theology” and “theologians” in what you wrote, and it reads just as well.

Which is exactly my point.

You can do philosophy with or without a rational analysis of objective observation, but science is nothing but. And, just as theologians routinely attempt to usurp the achievements of science for their own by declaring all science merely the exploration of the greater glory of the works of the favored pantheon, so, too, do philosophers take pride in the conceit that science is merely natural or applied philosophy.

Yes, yes. In ancient times science grew out of philosophy — but so, too, did astronomy grow out of astrology, and nobody would take seriously an astrologer’s claim that NASA is an astrological endeavor.

Again, we can make an objective observation as to activities that are philosophical and which are scientific, and it’s the rational analysis of objective observation where you’ll empirically find the dividing line.

Cheers,

b&

P.S. The thought that Spinoza is more important than Newton or Darwin or Curie or is…mind-bogglingly arrogant. Indeed, there’re

scoresof scientists more important — Ada Lovelace and Emmy Noether alone, if you just want to restrict it to those who did foundational work on theoretical frameworks. And if it’s something akin to “metaphysics” you think significant, that would be Laplace and his Daemon. b&>

Ben Goren,

You ask:

“How do we know which philosophical notions are and aren’t correct? Do we further philosophize about them, or do we go out and check the notions against the evidence?”

How do you know any particular observation or evidence proves anything? Maybe it just a one off, or million off result. You need philosophy to deal with that.

I see the problem as defining what knowledge is. We can have no certain or justified knowledge at all – we have conjecture and criticism. Knowledge consists of the conjectures that stand up to criticism (until they don’t). Even in what you want to call science, before you can even know what experimnents or observations to make, you first need a conjecture.

You also say:

“If you can point to a single philosophical notion deserving of serious consideration that is not supported by equally-serious observation, I’ll go out and buy a philosopher’s hat just so I can eat it.”

This is just a bad understanding of what philosophy is. Of course philosophers will welcome observations supporting there notions. They will need philosophical arguments to explain why the observations support their conjecture. For example, to offer a “serious philosophical notion” I’ll take “God does not exist” – this has withstood heavy criticism and is still standing, so I count it as knowledge, until the unlikely event some criticism actually stands up against it.

“Proof” in this context is a classic and incoherent philosophical superstition. You might as well substitute theological discourses on the meaning and purpose of life.

Centuries before the Caesars, Epicurus offered up the empirical observation that there are no powerful moral agents with the best interests of humanity at heart. As I reframe it, why don’t the gods ever call 9-1-1?

As it so happens, “god,” is, itself, an incoherent and logically inconsistent term. Yet, much of non-Newtonian physics is inconsistent with classically-understood ancient philosophical logic, so the incoherence should merely be taken as a strong indication that the phenomenon under consideration doesn’t exist, at least not in the form proposed. Our overwhelming confidence of the conclusion might start with the illogic, but the slam dunk comes from the neverending mountains of evidence contrary to any and all theological claims.

…to take but one more modern example, pretty much every modern deity is a creator deity philosophized into existence through the Aristotelian superstition that abhors infinite regress. “First Cause” is the common label. And when you compare philosophical and theological superstitions about the origins of the Universe with modern cosmology and multiverse theory…

…I mean, really? We’re taking those primitive, anti-scientific superstitions seriously…why, exactly…?

Cheers,

b&

>

I’m a philosopher who thinks metaphysics is extremely important, and Spinoza is one of the greats. But I also believe Emmy Noether has *also* contributed to metaphysics, by also discovering some extremely general truths about reality.

The problem is that “metaphysics” is itself as incoherent a concept as the divine and

primum movens.It presupposes that thereissome ultimately-fundamental nature whichcouldbe known.In stark contrast, ever since Turing and GÃ¶del, we’ve had reason to be overwhelmingly confident that such knowledge is impossible (you can never rule out a conspiracy) such that even the very concept is incoherent.

You could, after all, be a brain in a vat, and the vat could be a subroutine of the Matrix, which could be a program running on the Holodeck, which could be the subject of one of Alice’s Red King’s Dreams, which could be an hallucination induced in a wino by the CIA using Martian mind-ray technology.

What we can do is learn all we can about reality as we observe it. And, even should some conspiracy theory hold, we might even be able to gain some knowledge of it — but not if the conspiracy is “good enough.” But to go beyond that is to engage in useless theology. Your search for “metaphysics” is a search to uncover the ultimate conspiracy underlying all other conspiracies…and, by its very nature, your search can’t even hypothetically involve investigation of observation.

If you

dowant to gain some understanding of reality as it presents itself, including questions of origins…well, the cosmologists arewayahead of the philosophers.We have good reason to have some, not much but some, confidence in modern multiverse theories of cosmogenesis. And we have absolutely zero reason to take seriously any proposals for ontologies more fundamental than that — at least, not until we have good reason for very strong confidence in an explanation for the horizons we’re currently probing.

Cheers,

b&

>

Well, I never metaphysical I didn’t like.

Keith Douglas,

“I’m a philosopher who thinks metaphysics is extremely important, and Spinoza is one of the greats. But I also believe Emmy Noether has *also* contributed to metaphysics, by also discovering some extremely general truths about reality.”

I remember Noether (very vaguely) from grad school algebra as being involved with Ring Theory. Is she the same?

I see Spinoza as the key historical figure in bringing free, secular governments into the world. He wasn’t alone in this, but I would rank him highest.

Carl: Noether proved that for every conservation law there is a corresponding symmetry in (the property of) things. I know she did work in algebra as well, so your remark about ring theory makes sense.

Yes — and the scientific principles we have the most confidence in, the ones that have never demonstrated even the slightest hint of inconsistency nor unreliability, are those about conservation. That’s a big part of the attraction of Lawrence Krauss’s “Universe from Nothing”: the total energy of the Universe is zero, so everything is conserved.

(One might wonder how energy can be zero but still lots of stuff driven by energy can happen. The answer is that stuff isn’t actually driven by energy, but by entropy…but that’s a topic for another day.)

Thanks to Noether, all of physics can succinctly, poetically, and accurately be described as a search for symmetry. Find the symmetries and how they break and you’ll understand what’s going on.

In stark contrast, from Spinoza’s metaphysics, we get nonsense such as this:

Pure Aristotle, the special pleading of an unmoved mover to stop that evil infinite regress. We’ve no sign that there actually

isanything fundamental, and there isn’t even any way in principle that you could determine whether or not whatever you think is fundamental actually is. Even now, with the completion of the Standard Model thanks to the discovery of the Higgs by CERN’s LHC, we’re highly confident that there’s still physics beyond the Standard Model to be found — meaning that the bosons and fermions of the Standard Model aren’t themselves fundamental Spinozan “supstance.”Pure dualism. The Universe gives not one whit what we do or don’t think of anything, and our perceptions influence reality not at all. (At least, no differently from any other physical process.)

And he ties it together thusly:

Seriously? That may well have been fine and dandy a few centuries ago, but today?

And his philosophy of mind is…incoherent. “The human Mind is a part of the infinite intellect of God” and similar such nonsense. We’re talking Chopra levels of woo.

Whatever influence he might have had on earlier scientists, there’s absolutely nothing of Spinoza in modern science in any meaningful form.

Cheers,

b&

>

Ben Goren,

Spinoza may sound like Aristotle, but only because the vocabulary of Aristotle was what he had to work with. What Spinoza did (continuing what Descartes and others began), was break the hold Aristotle and Christianity had on Europe for preceding centuries after the church had nearly erased all trace of Epicurus. You can see it as an Epicurean revival.

The quotations you cite sound damning, but in the context of his full argument they make eminent sense (unless you are a theist). In modern terms, Spinoza sets out to say there is nothing outside the universe, nothing supernatural, and no god. Knowledge does not come from authority (church, Bible, or Aristotle), it comes from good explanations.

Spinoza’s most important contributions (in my view) are in political philosophy – his influence on modern liberty and modern government. The neuroscientist Antonio Damasio has written “Looking for Spinoza: Joy, Sorrow, and the Feeling Brain” praising the philosopher’s relevance to what he is discovering in his lab. Spinoza is widely recognized as the founder of modern Bible criticism, employing historical, literary, and linguistic techniques used by religious and secular scholars alike – even most religious scholars now must agree that the Bible is a thoroughly human book. Imagine if the Islamic world had this view about the Koran, or the West still held the pre-Spinoza view.

Most influence Spinoza has on cutting edge science, will have to be the sort he had on Einstein, which in Einstein’s own estimation was profound.

Ben, a serious question. You are obviously very smart and very well read. What have you read of or by Spinoza? How did you form your opinions about him?

You write after quoting Spinoza’s definition of “attribute:”

Pure dualism. The Universe gives not one whit what we do or don’t think of anything, and our perceptions influence reality not at all. (At least, no differently from any other physical process.

Spinoza is generally considered a monist, the exact opposite of a dualist. The rest of your words here are pure Spinoza, and not in a peripheral manner. He says exactly this, if not verbatim, clearly enough no one would find a difference.

Honestly? Not much.

But I haven’t read much of Aquinas, either. I ate up Lewis’s

Narniaseries as a child, but haven’t read much of his straight-up apologetics. My reading of other “serious” theologians is equally slim.And for the same reason. When the go-to money quotes in Wikipedia and similar sources are so bad, and when those quotes match the overall picture painted…what’s the point? Life’s far too short to waste on such nonsense.

And, even more to the point, what matters is not who said it, but what is said. It takes me conscious effort to remember that Eratosthenes was the one who came up with the sieve for finding prime numbers — but I used to use that algorithm as an assignment for an introductory programming class I taught many years ago. So I really don’t care that it’s Spinoza who thinks that our minds are parts of an overarching divine supermind; such superstition is long-since-debunked nonsense right up there with the Philosopher’s Stone.

I note that, in your defenses of Spinoza’s importance to science, you’ve yet to point to any concrete facts about nature which he put forth. In stark contrast, I gave you multiple examples such as Noether, who identified the connection between conservation and symmetry, which is the very essence of all modern physics. Or Laplace and his Daemon, a paradigm that has held fast ever since. (“Give me the complete current state of the system and all the rules by which it functions, and I’ll give you all past and future states.”) I could keep going — Heisenberg who identified the inherent limits of resolution, Einstein who demonstrated the lack of absolute reference frames, and so on.

If you really want to convince me of how essential Spinoza was, you could offer up some such examples…

…but I think we both know they’re not forthcoming, else they’d be as much at the tip of everybody’s tongue as Newton and Mechanics or Darwin and Evolution.

Cheers,

b&

>

OK, I’ll drop it. I don’t want to irritate you. If you think Spinoza belongs with Aquinas and C.S. Lewis, you just lack information.

It’s not merely that I equate theologians and philosophers. It’s that apologists for both equally fail to produce evidence supportive of their claims — merely assertion and argument, but no evidence.

Cheers,

b&

>

Claim: Spinoza “broke” the Bible. He undercut theistic religion by demonstrating, the Bible was merely human literature. I hope I don’t have to give evidence that this was important to the scientific and political revolutions that followed.

Evidence: Spinoza’s Theological-Political Treatise, using linguistic, textual, and historical evidence, has now gained wide acceptance – both his conclusions and his methods, even by theologians and Bible Scholars.

Okay, that’s well and good. But it’s religion and politics, purely human endeavors, and not even remotely hypothetically tangentially related to our understanding of reality.

It’s also far from new. Anybody throughout history who wasn’t an Abrahamist could have told anybody who asked that the Bible was merely human literature.

You’d probably argue that Jefferson was inspired by Spinoza and so Spinoza gets all the credit, but I’d argue that Jefferson and his “Wall of Separation” did

farmore than Spinoza to free the West from the shackles of Christianity. And let’s not forget the Jefferson Bible, which is much better known today than any of Spinoza’s works.Cheers,

b&

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Please see post #39 for my reply.

“maybe the true surprise is that Pythagorean triangles are expressible rationally….”

I suppose, if you’re fiddling around adding up pairs of squares, you’re likely going to come across a few cases where they add to make another square, just on the odds. On that assumption, it would be odd if there were no Pythagorean triangles.

cr

Keep forgetting to ✔️✔️

Yes, the set of rational numbers has measure zero in the real line, so it is unsurprising that pi is irrational, and in fact (as pointed out above), transcendental.

In fact, if one were to just randomly pick out any number in the real line, the probability that the said number would be irrational is 1.

This whole question just demonstrates our prejudice toward the rationals.

The fact that e^iπ +1=0 is perhaps the best argument for the existence of god. But no god could be that beautiful

Ah, Euler’s proof.

If I remember the story, as Asimov to.d it, iirc, Euler was debating with an atheist, who had prepared compelling philosophical arguments for God’s non-existence, but knew no math[s], so was too flummoxed by Euler’s proof to continue.

Kind of like WLC (until he came up against Sean Carroll).

/@

My favorite.

No idea, but it is cool that we know that it is, rather than not. As an experimental physicists, it’s irrelevant to me. I can define 2 as 2.00000…. with an infinite number of zeros and it is now as long as any irrational number, including pi. To me, all numbers are the same. And 1/0 = infinity (for any rigid math teachers out there).

Why are there three quarks to a proton? Why is the lifetime of 22Na about 2.6 years? These questions may have specific coherent reasons, but all the answers are ultimately tied to some arbitrary way in which nature is.

It’s just the way that it is.

“I can define 2 as 2.00000…. with an infinite number of zeros and it is now as long as any irrational number..”

Right, for sure.

But you could even have defined 2 to be 1.999999…., in case somebody quibbled that 0’s are not fair. Or by using base 3, as 1.222222…., as well as 2.000000… of course. So your ‘unfair’ opponent has nothing left to buttress their argument.

In any case, further above several people are a bit misleading if they are implying at all convincingly, either via Cantor’s cardinals, or even via measure theory I think, that \pi is irrational because the probability is zero that a ‘randomly’ chosen number is rational.

One reason is that \pi is computable and the set of computable numbers is merely countable. (After all, you can easily ‘infinitely list’ all programs and so all which happen to run forever spitting out a digit every so often.) And I think it is again merely a set of measure zero, but haven’t checked, so that puts doubt on the measure argument as well.

Additionally, as David Wallace has cogently argued in his long Born rule section in the book on Everett’s quantum many worlds, a convincing meaning of the word “probability” is still very far from clear to either scientists or philosophers. And if anything that is easier to make progress on in the Everett multiverse.

Finally to another above, I’d argue for ‘e’ as being the more basic than \pi in the fame contest for which transcendental number is the winner. My reason is that ‘e’ arises from a 1st order DE (differential equation), namely y’=y,

whereas \pi from a 2nd order, namely y”=-y.

Sorry, countable implies measure zero, so I didn’t need to check. I’m a dope there!

There are *many* more irrational numbers than rational numbers. One way to look at this is that if you look at the rational numbers in the interval from 0 to 1, they take up, mathematically speaking, zero amount of space. They’re countable, i.e., they can be put in one to one correspondence with the positive integers, while the irrational numbers, as Cantor famously proved, are uncountable.

So it would actually be surprising (and mean something) for an arbitrarily chosen number to be an integer, or even rational.

Pi (and e) are more than irrational – they’re also transcendental. That means they’re not roots of a polynomial with rational coefficients. The square root of two, again famously, is not rational. (You can prove that by a wonderful technique called infinite descent.) But it’s not transcendental: it’s a root of the equation x^2 – 2 = 0

Granted that an arbitrarily chosen number is overwhelmingly likely to be irrational. But doesn’t that just beg the question of whether pi counts as an arbitrarily chosen number?

Did you know that if you raise e to the power of pi and then subtract pi, you’ll get exactly 20? It’s a good way to check how well a computer program or spreadsheet handles high-precision arithmetic.

By the way, that was a joke.

Just as well you mentioned it. 🙂

See also

https://xkcd.com/1047/

cr

Possibly trivial semantic observation, but my mnemonic for recalling the meaning of irrational numbers is to remember that the opposite of irrational is rational and that rational numbers are those that can be written as the

ratio(instead of saying fraction, even though they are effectively the same) of two integers. An irrational number cannot be written as the ratio of two integers. For example, 3 is a rational number because it can be represented as the ratio 18/6.Well, yes, that’s exactly the etymology of the terms.

Etymology is often helpful in understanding words. Unlike entomology …

/@

Unless the word you are trying to understand is “Drosophila”.

I have a somewhat different take. Pi is not just an abstract property of Euclidean geometry. It shows up in quantum mechanics.

The fact that it is transcendental suggests to me that reality is analog rather than digital. In other words, not the matrix. Reality is not countable.

Just my favorite woo.

I agree with that woo.

Way-back-when in grad school I decided that integers were artificial constructs (and therefore so is the concept of “irrational”) and that mathematics in general doesn’t jibe well with reality because the closer mathematics came to describing reality the more complicated it got, e.g. Newtonian vs Relativistic physics.

Like a smarter guy said: “As far as the laws of mathematics refer to reality, they are not certain, and as far as they are certain, they do not refer to reality.”

BTW – one on the most amazing, yet simple equations in mathematics is Euler’s equation:

e^(pi*i) = -1

(Looks cooler with superscripts and math symbols!)

That equation puts together two transcendental numbers and one imaginary number, and gives you back an integer.

Geometrically, it says that if you take 1 (thinking of it as lying on the x-axis in the plane) and rotate it 180 degrees around the origin (Pi radians), you end up at -1.

That equation has no right to exist!

It’s just too damned unlikely. And bizarre.

cr

Better written as e^(pi*i)+ 1 = 0, to get in two other greatly important numbers.

There is an infinite number of rational numbers, but I suspect there are a good number of irrational mathematicians.

BTW, the fact the Pi is the ratio of the circumference to the diameter of a circle is lurking around in the back of that proof, though it takes a bit of looking to find it. I has to do with how the function sin(x) is defined and the units used for the function (radians).

Ain’t math fun?

I’m quite content that π is irrational given circular geometry. The really astonishing thing is that it turns up in so many other places.

/@

The answer to Jerry’s question is that there is a difference between proving that π (the mathematical constant) is irrational (transcendental) and explaining _why_ the ratio of the circumference of “standard” circle to its diameter is that number. Just as he wrote, a geometric step is required to link these.

Someone already said that in a curved space the ratio is not π. A nice example is the sphere. If you are limited to two dimensions and measure distance along the surface, then the diameter of the equator consists of two half of meridians. The ratio C/D is then exactly 2.

What we commonly mean by the circle is a curve equidistant from a point on a plane with Euclidean metric (which roughly means flat space). It then so happens that the position of all points on the circle can be given by (x,y) = (r cos(t), r sin(t)), r being the radius and t going from 0 to 2π – precisely the mathematical constant, which must be the period of the trigonometric functions. (Note this is not the only possible parametrisation.)

The integral formula for the length of such curve uses the derivatives of the above and is C = ∫sqrt(r^2 sin^2 t+ r^2 cos^2 t)dt = 2πr, thanks to the known trigonometric identity.

I would have to sit down and try to write it but if pi were rational, then a circle would have to be a polygon, wouldn’t it?

For example, if pi were 3, then a circle and the inscribed hexagon would have the same perimeter, which would mean they are identical.

I am fairly sure for any rational number p < pi there is an inscribed polygon that has perimeter r*2*p … since all inscribed polygons perimeter is supposed to be smaller than the circle's …

I think that is it exactly. If 22/7 were right, you could have a 22-agon shape with a “diameter” of 7. The same would be true of any other ratio. Therefore pi must be irrational.

Part 2:

The conjecture is true if you allow irregular polygons (and insist p > 0). A proof could go like this. Pick p r*2*p. Fix one vertex and slide all the other vertices of the n-gon along the circle toward the fixed vertex. The perimeter of the n-gon with slid vertices will shrink continuously toward 0 (picture all n vertices almost coinciding). The continuous shrinkage from q to 0 insures at some point the perimeter will equal r*2*p, since q > r*2*p > 0.

Great to see so much math interest here.

The conjecture

“for any rational number p 0). A proof could go like this. Pick p r*2*p. Fix one vertex and slide all the other vertices of the n-gon along the circle toward the fixed vertex. The perimeter of the n-gon with slid vertices will shrink continuously toward 0 (picture all n vertices almost coinciding). The continuous shrinkage from q to 0 insures at some point the perimeter will equal r*2*p, since q > r*2*p > 0.

Please disregard this, I had trouble posting.

Great to see so much math interest here.

The conjecture

for any rational number p 0). A proof could go like this. Pick p r*2*p. Fix one vertex and slide all the other vertices of the n-gon along the circle toward the fixed vertex. The perimeter of the n-gon with slid vertices will shrink continuously toward 0 (picture all n vertices almost coinciding). The continuous shrinkage from q to 0 insures at some point the perimeter will equal r*2*p, since q > r*2*p > 0.

Please disregard this, I had trouble posting.

For some reason, what I’m trying to post keeps getting the middle cut out. I’ll try splitting it in two.

Part 1:

Great to see so much math interest here.

The conjecture

“for any rational number p < pi there is an inscribed polygon that has perimeter r*2*p … since all inscribed polygons perimeter is supposed to be smaller than the circle's"

is false if you stick to regular (all sides equal length) polygons:

An n-sided polygon will have some perimeter P1 strictly smaller than P2 the perimeter of an n+1 sided polygon. The infinite number of rationals between P1 and P2 will never be the the perimeter of a regular polygon.

Well, it works if you allow for open inscribed polygons, right?

So, for any fraction p/q < 2pi, you can build a polygon (open or closed) where all vertices lie on the circle of radius 1, and the length of all the sides is exactly p/q.

Since it won't be a circle, because distance to the center is not constant, it shows no p/q < 2pi works, therefore pi can't be expressed as p/q?

This is how Archimedes calculates the value – he shows that 3 10/71 < pi < 3 1/7

See W. Dunham, _Journey Through Genius_ for a modern version.

God is the value of the last significant figure of pi.

I haven’t watched the video yet, but here’s my guess.

Pi is just a special case of calculating the length of a curve given by some function f(x). We can do this by taking the first derivative of f, applying the Pythagorean theorem, and integrating sqrt(f'(x)^2 + dx^2). In effect we’re adding up an infinite sum of infinitesimal hypotenuses.

Since this involves taking square roots, we should expect the result to be irrational, except in special cases constructed around Pythagorean triplets.

Any real mathematicians out there should feel free to poke holes in this argument.

Why is pi irrational … a serotonin imbalance?

Maybe Ω and λ can pitch in and get pi’s script refilled …

i said to pi “Be rational.” Pi replied to i “Get real.”

“And I’m a bit surprised that nobody uses the irrationality of π as an inexplicable fact about mathematics that implies the existence of God.”

In the book version of the 1997 scifi movie Contact, (not in the movie) something like that was at the end: It was discovered way… way… way out in the decimal part of pi the numbers started a sequence or code that would plot up as geometric shapes (or something like that). It was implied that it was the “signature” of the creator, and purposely built into the geometry of the universe: not necessarily god, but an advanced civilization that was creating universes through big bangs.

You beat me to it.

It’s explained here -> here and “Since pi is infinite, do its digits contain all finite sequences of numbers?” (Answer: maybe).

It was implied that it was the “signature” of the creator, and purposely built into the geometry of the universe:From the second link above:

++

The problem is that pi is not a physical constant, it is a mathematical constant, the ratio of the circumeference of a Euclidean circle to its diameter. It is not adjustable because its definition makes no reference to reality. Furthermore, there is no distinct physical constant that would be meaningful in any universe even vaguely resembling ours. In curved spaces, there is no constant ratio of circumferences to their corresponding diameters. The small circle limit, where no singularity is involved, is precisely pi, hence the qualification “distinct”.

++

Useless toy #437 (using the same numbering scheme as for useless facts), the first occurrence of my (UK, mobile) phone number in π is at position 52216852 after the decimal point.

You can go one way, but not the other, with http://www.angio.net/pi/ and probably other tools. Unfortunately, it won’t quite stretch the 3 extra digits necessary to give the number fully, because it only looks at 200 million digits of π .

Why is Pi irrational?

‘cos God hates mathematicians?

cr

FTFYThe FSM (Sauce, etc) has some really neat stuff lined up for mathematicians, but this margin is tooo0″£$^£&$ NO CARRIER

There are some very sharp people commenting here so I will ask a question that bothers me occasionally. I think I read somewhere that (A) the decimal digits of pi are proven to be a random sequence which (B) implies any arbitrary sequence of digits (however long) *will* appear (eventually) in pi (as expressed in decimal digits).

I would love to know whether either (A) or (B) are true, and especially whether (A) and (B) is true.

If (A) and (B) are true, I suspect this is one of those things in which theory and practice don’t intersect because of the practical difficulty you’d need untold googols of universe lifetimes to find the billion digit sequence (which encodes a high def picture of Hili the cat say) within the digits of pi.

(A) It depends on what you mean by “random”. In fact, I would say that random is probably not the best word in any context. It might be more appropriate to say the digits are “arbitrary”.

(B) is something *much* stronger. It is certainly not true that (A) implies (B) in general; not for pi, not for sequences generated by sampling digits 0-9 uniformly at random. It is strongly suspected that pi has this property, but it is not known.

A) >It depends on what you mean by “random”.

Indeed, it is difficult to express exactly what I mean because clearly the digits are absolutely not random – they’re the digits of pi! I wish I had my original source reference – I would speculate that it would be something like “the digit stream specifies to arbitrary accuracy the mathematical constant pi but in all other respects has every known property of a stream of random digits”.

B) I am afraid I don’t follow you here. Are you saying that an infinite stream of random digits does not necessarily include any arbitrary finite stream of digits? That would be disappointing. Remember that infinity is a lot :- ) Also when you say it is strongly suspected that pi has this property, but it is not known, which property are you referring to?

There’s another property, called “normality”, IIRC, which means that any finite sequence of digits also exists. I do not know whether it has been proved that pi is normal.

(Why are we doing this all now? The next science holiday is mole day, not pi day!!)

Yes, but pis taste better than moles.

Thank you, “normality” turns out to be the key concept here. Some googling turns up the information that pi is not proven to be normal, although it is strongly conjectured to be so. The money quote (in Wikipedia) is ‘In particular, the popular claim “every string of numbers eventually occurs in π” is not known to be true.’ As far as I can tell this is equivalent to restating that pi is not (known to be) normal. Basically normal means the digits (in any base) are randomly distributed. It *is* known that almost all real numbers are normal, so almost all real numbers do have a HD picture (video even) of Hili the cat in there somewhere, if you just go far enough.

Within the context of infinity, pi could contain the sequence of digits in e and e could contain the sequence of digits in pi.

Damn, my head just exploded and now someone (else) is going to have mop all the gore off the screen and desk. Or are they. Can that really be true? I know there are different types of infinity, some stronger than others – but these are the same types of infinity. Clearly finite can be a subset of infinite. And I can believe weak infinite can be a subset of infinite. But how can peer infinite be a subset of infinite? I’m not even going to try and think about the recursive aspect of your comment. As Clint Eastwood would say “A man’s got to recognise his limitations”.

There was a project some years ago which used some fancy maths to calculate the (IIRC) quadrillionth (binary) digit of π

withoutcalculating the previous [lots of] digits : pihex. It was a distributed computing project that used 1900-ish computers from a little shy of 700 workers (of which I was one, contributing two computers when I was onshore). In particular, it was carefully designed to not need internet access unless a new work unit was requested manually, which was just as well since it was before I had regular internet access.Given that such weird maths is possible, I wouldn’t even touch a bet that it was impossible to find an arbitrarily long (binary) string in π , though the calculation may take some time and require substantial intermediate storage.

Dare I mention Mandelbrots? A graphical representation of the weirdness produced by digging deep into an algorithm.

What fascinates me is that you can take a good computer program and zoom in and in to an arbitrary point and what you see has probably never been seen by anybody else ever and may never be. Even though anybody else navigating to that exact same point under the same conditions would see precisely the same thing. But there is so much complex detail, I think the likelihood of anyone else ending up at that precise location is vanishingly small.

(I suppose it’s a bit like picking up a grain of sand on a beach. The odds of anyone else picking up that exact grain of sand and examining it are minuscule).

cr

Having seen a few beaches excavated and exported to Saudi Arabia, I think the odds of the sand grain having been handled before are better than those for finding a previously-viewed grain of the Mandlebrot set.

I did program a Mandlebrot viewer some years ago. Even without having access to double-precision numbers (this was BASIC, after all), I was unable to find limits to the depth I could get to.

I was a little bit careful to say ‘picking it up and examining it’. I originally said ‘seeing’ but, standing on top of a cliff, I could technically ‘see’ every grain on the surface of the beach.

But I do agree with you, I think the Mandelbrot set is (probably) potentially infinite in its recursion. I’m not so sure about actual Mandelbrot computer programs, I think it eventually reaches a limit. In ‘Xaos’ for example, I seem to have reached a limit at a radius (view diameter) of 5e-14.

cr

P.S. I did write a program for Mandelbrots, in BBC Basic 5 for the Archimedes. That language was almost uniquely well-suited to Mandelbrots, since it had a built-in assembler that could be called from within the program. So one could write the routines for selecting the screen area, display etc in easy-to-program Basic, and the short but multiply-recursing (many hundreds of recursions per point) calculation loop in much faster assembler.

cr

That’s a factor of 10^18 off from the Plank length. Does this program (hang on – just Synaptic-ing it) even manage to see the universe it exists in?

I’m too lazy to go hunt down the source code and try and follow what it does.

But good luck.

cr

Apple pi is irrational, too. Why else ditch the headphone jack?

Cherry Pi-e is irrational also, I broke a tooth on a cherry pi-t a couple of decades ago.L LMFAO—I wasn’t laughing the day it happened though.

Pi has lost its shimmer for me over the years, jaded by grad school perhaps. What fascinates me though is that irrational numbers are so damn hard to find in practice, yet they are everywhere in theory. As several commenters mentioned above, there are far more irrationals than rationals (in terms of set cardinality); you can say that there are as many irrationals as real numbers, but only as many rationals as integers. Yet it is notoriously difficult to find most of those irrationals.

Sure, we all know about the various algebraic irrationals (various roots of integers), but this is an extremely small set – also only as big as the set of integers. Where are all those other irrationals? We see pi and the Euler constant e all the time. And you’ll see some other less common ones like phi for the golden ratio. But that’s about it. Pi pops up everywhere because of its intimacy with the circle, and e pops up everywhere because of its closeness with the exponential (limit of (1+1/n)^n for example). But write down a closed form expression for me that defines an irrational number that isn’t just a rational scaling of one of the common ones. Extremely difficult to do.

Also, the irrationals aren’t closed in any coherent way, which can lead to all sorts of funky things like the fact that there exist irrationals a and b such that a^b is actually rational. Or what about the fact that we don’t even know if pi + e is irrational?

It’s no surprise to me that many mathematicians are deeply religious. I met many of them as a grad student. It’s a beautiful subject, but its beauty can be absolutely blinding.

“..many mathematicians are deeply religious.”

Any real evidence for that?

At least with respect to outside of USians, I very seriously doubt that. And though it might be hard to get the evidence, I’d be willing to bet that, within the U.S. National Academy of Sciences, the mathematicians are no more religious than the rest, religiosity being quite rare there.

There seems to me to be a kind of extreme dislike of anything slightly like Platonism, pretty shallow I’d say in many cases. Then the logical mistake is made that ‘belief in existence of something outside the natural world implies belief in some kind of god’, whereas it’s the converse of that which is pretty obvious. That quoted belief, which is nonsense, then combined with the fact that many mathematicians adopt a Platonist philosophy of mathematics, leads to acceptance of the quote at the top.

It is interesting here that many refer quite straightforwardly to irrationals, to cardinality, to measure theory. I wonder how many, if not platonists, have thought much about what they mean by those words (except Ben Goren, who I’m quite sure has thought seriously about that.)

I sense you have taken a bit of offense.

That statement was meant to be taken relatively; i.e. many mathematicians are religious relative to the proportions who are religious in the other sciences.

My evidence is largely anecdotal, although I do recall seeing several surveys that polled religious belief among different scientific fields. Mathematicians were always up in the highest category of religious belief. I do not recall sources though, so you can either try to Google them or not believe me.

I don’t appreciate the implication that I don’t know what I’m talking about. You could very easily click on my name and go to my website, which would corroborate the fact that I’ve spent plenty of time thinking about measure theory and the like.

No offense, just asking for evidence. If Googling for surveys will produce such evidence, perhaps it’s the asserter of that claim, viz.

“It’s no surprise to me that many mathematicians are deeply religious. “,

who would be kind enough to find it, not the person (me) who has expressed skepticism.

I will say however that my last two paragraphs previous didn’t, and were not intended to, refer specifically to you. I have no idea what position you take, platonish or not, with respect to a philosophy of mathematics.

“But if there’s some proof out there that the ratio of a circle’s circumference to its diameter, based on the geometry alone, must be irrational, I’d like to know about it.”

Not a proof, but I have thought about this so I will share my layman reasoning.

If you describe a circle as a polygon with infinite sides then it kinda makes since that pi is an irrational number when the number of sides cannot even be expressed as a number.

I was thinking on those lines myself.

Once you get above a certain integer “

n“, then ann-gon can be inscribed within a circle, while an (n+p)-gon (pis also an integer, and might be 1 or higher ; if it’s 1, you get as tight a fitting of inscribed-gon, circle and escribed-gon as is possible ; if it’s 2, then you don’t have to worry about even integers in your ratio)would be escribed by the same circle (call these the in-gon and e-gon).If you sketch the relations between the in-gon, the circle and the e-gon, you can see that by adding 1 side to the in-gon and 1 (or more) side to the e-gon, you’d get a better fit between the three figures. (I’m pretty sure I’m stealing this from Archimedes’ method for calculating a value for π so I’m feeling pretty good on sketching the proof this far.)

Our in-gon and e-gon represent the numerator and denominator in a

rationalapproximation to πBy adding to the in-gon and the e-gon, we get a better approximation to π involving larger numbers.

I can’t see any way that you can stop this process of improving the fit, for any

n, so I can’t see that any ratio of integers can produce a rational approximation to π whichcannotbe improved by this process. And I think that’s a mathematical hop, skip and a jump from “therefore π is irrational, QED.”I could probably tighten that up a lot before I run out of maths, but I’d need to learn HTML maths mark-up far better than I know it now to express that, and I’d probably break WordPress’ rendering engine in the process.

Yes, the ratio of the perimeter to the long diagonal of a regular n-sided polygon approaches pi as n increases. Incrementing n increase the precision so the digits cannot terminate and the sides and diameter of the polygon change so the digits cannot end with an infinitely repeated sequence (waves hands.) Ergo, irrational.

Alldiagonals, not just the longest (nor the shortest) approach π.That’s another way of looking at it. Were π rational, that would imply that all arbitrary diagonals of all polygons are also rational, which would further imply that all numbers are rational.

b&

>

I’m missing the argument here. A sequence containing irrationals can have a limit which is rational; for example, sequence with nth term being 1/(square root of nth prime) has the rational number 0 as limit.

So you seemed to me to be saying a rational limit for a sequence implies terms of the sequence are also rational. But it’s late, and maybe I’m just missing the point.

“Pi is irrational” is a mathematical truth. Like all provable mathematical truths or theorems, you make definitions and axioms then by deduction prove the theorem. All provable mathematical truths are of the same nature as “there are 36 inches in a yard” – they are in a very real sense just tautologies. Some are just more complicated to see than others.

I’m using “provable” restriction above, because Godel has shown there infinitely many unprovable truths in any sufficiently complex mathematical system.

“And then the astute theomathematician could bring up the square root of two. . .”

I’m late to this, but still surprised no one ran with this a bit more. There’s the root of -1.

I’m convinced there is a really good joke in the window/bumper stickers I’ve seen lately of “HE>i” from the clothing company hegreaterthani.com – them saying its based from John 3:30. “He must increase, but I must decrease.”

Wouldn’t it be more accurate simply to say:

HE = i

i = -1^0.5

HE = imaginary

If I were just to throw up my own sticker with “HE=i” people would just think I was boasting and wouldn’t get the joke. That’s why I think it’d be better as a proof. But still one more subtle than the above that gets the observer to conclude the imaginary part.

There is something funny there. I just can’t write it right.

If you think I’m claiming Spinoza is *exclusively* responsible for so many good things, I am not. Spinoza is part of a radical tradition, including (not exhaustively) Epicurus, Lucretius, Machiavelli, Bruno, Descartes, Hobbes, Locke, Hume, Franklin, Paine, and Jefferson – who all played important roles in creating the modern world. As did Galileo, Newton, and many others.

The foremost thing done by these radicals was to discredit the idea there is some authority in regards to knowledge, which pre-Enlightenment was Aristotle and the Bible. They broke this stranglehold. The basic idea that we live in a natural, explainable world emerged – where science and evidence was more and more valued.

You write:

“It’s also far from new. Anybody throughout history who wasn’t an Abrahamist could have told anybody who asked that the Bible was merely human literature.”

If it hasn’t been clear from context, I’m talking about the West. Spinoza was *the first* to publish this claim, and as seems to be important to you, give copious evidence.

As to Jefferson, not only did his Library contain the works of Spinoza, he lived in an intellectual environment saturated by Spinoza. Spinoza, of course, doesn’t “get all the credit” but if you know the history he deserves a great deal, as do many others: Bolingbroke, Sidney, Chubb, Shaftsberry, Toland, Locke. Almost all largely forgotten, except Locke. You have admitted you know little about Spinoza, so why do you continue to make statements about him? I doubt if you are interested, but others might find this list of books useful:

By the man himself:

Ethics (1677): (avoid the Elwes translation, look for something done later like Curley or Shirley): This book is notoriously difficult and hardly anyone will jump into it and comprehend much.

Theological-Political Treatise (1670): Straightforward reading demolishes the Bible, miracles, prophets, the supernatural in general. Provides cogent explanation of why free speech and thought are so important. This also has the honor of being included among Jerry Coyne’s short list of atheist books: https://whyevolutionistrue.wordpress.com/2014/03/20/i-got-the-book-you-know-the-one-with-the-best-arguments-for-god/

Steven Nadler (2011):

A Book Forged in Hell, Spinoza’s Scandalous Treatise and the Birth of the Secular Age.

A very readable, entertaining look at the TTP, the circumstances and immediate aftermath of its writing.

Rebecca Goldstein (2009):

Betraying Spinoza, The Renegade Jew Who Gave Us Modernity. Very readable, gives an overall picture of Spinoza (by a philosopher who had a Ben-like attitude until about the third time she had to teach a Spinoza course), along with some speculation by a top novelist.

Matthew Stewart (2014):

Natures God, The Heretical Origins of the American Republic.

My favorite book. It gives a rigorous argument that America is not a “Christian Nation.” Well researched and documented – one third of the book consists of end notes. It focuses on the Declaration of Independence and the “radical” philosophers I’ve mentioned.

Jonathan Israel:

Radical Enlightenment (2001)

Enlightenment Contested (2006)

Democratic Enlightenment (2011)

You have to be pretty serious to tread here. Three thousand-page books by a historian digging through evidence from the previously unknown or obscure through to common knowledge.

If anyone is still reading, and it isn’t clear, this is a reply to Ben Goren’s last post.

Just to put things in perspective, you started this diversion by claiming that Spinoza was a more significant figure than any scientist. Now you’re including him as a minor standout of many peers in a long laundry list of antiestablishment figures. And your evidence largely consists of a list of works that you yourself describe as “notoriously difficult” and borderline incomprehensible.

You’ll excuse me if I’m decidedly underwhelmed by this retraction, and also if I’m struck by the obvious parallels with Christian apologists insisting I haven’t correctly read the right theologian to properly appreciate the true sophistication of their arguments.

Cheers,

b&

>

I made no retraction. I stand by the claim that Spinoza was the most important figure. You transmogrified my claim to something like “the only important figure” and I was pointing out my claim was not so absurd. And you misunderstand me if you think I’m saying he was a “minor standout of many peers” – I think he was the central figure in that list of greats, the one who crystallized what came before and deeply influenced and set the tone for those who came after, along with much of western culture.

That’s my view, Neil deGrasse Tyson thinks it was Newton, and many probably would pick Darwin. Good choices, but I disagree. Maybe I could be convinced to change my mind, but this opinion has been jelling for 50 years, so it’s unlikely.

I’m not so much arguing you haven’t read the right thing, which is not in itself an invalid argument, I’m pointing out what you have already admitted, you have read almost nothing (about Spinoza). It’s not the sophistication of any argument holding you back, it’s the utter lack of knowledge concerning the topic (Spinoza, his achievements, his views, and his influence). It’s well beyond my powers to convince anyone, using a few hundred words, of such an admittedly bold claim, particularly someone with only misconceptions about the subject to begin with. Hence the book list.

There is one book on my list I would class notoriously difficult – Ethics. The Israel trilogy is extremely long and detailed, and no doubt most will find it boring. The other four sparkle. I could add many additional titles.

You’re certainly entitled to your opinion, but the facts are that his physics are entirely unrelated to reality and his theology and ethics substantively no different from positions expressed at least two millennia earlier by Epicurus and his followers. As Stalin would have noted, he had no divisions. As you yourself have noted, his most important work is obscure and nigh on incomprehensible.

So, at most, though he had no intellectual innovations to his name, he revived and revised some significant ancient ideas and got a lot of other stuff very, very worng whilst expressed himself badly.

In stark contrast, Newton invented (independently at the same time as Leibniz) the most-practically-applicable (still to this day!) field of mathematics, and used that invention to almost completely describe the physics that’s most overwhelmingly relevant to human scales. Darwin showed how complexly organized life can (and does) spontaneously evolve from simple disorder — thereby slamming shut the last remaining door for the gods. Louis Pasteur is directly responsible for pasteurization and vaccination, which has saved literally billions of lives, including yours. Edward Teller rewrote the geopolitical map with the fusion bomb. Kennedy put Armstrong on the moon, and Kennedy and Khrushchev brought civilization to and back from the brink of self-immolation. Stalin forged the Soviet Union. Genghis Khan grandfathered innumerable great-great-grandchildren at the same time as he initially wrote the first geopolitical map. Caesar crossed the Rubicon. Paul popularized (to a rather limited extent) an obscure ancient Jewish demigod who most recently had been no more than a footnote in Philo’s theology, thereby setting forth the roots of Christianity — and the author of Mark gave Jesus his biography.

If, amidst all those facts, you still think that a revivalist who had no clue about how reality functions was the most important figure in history…well, that’s an opinion you’re certainly welcome to. But it says far more about what’s important to you personally than to any sort of influence the man actually had on history.

Cheers,

b&

>

You do a lot better better for yourself by proposing alternatives. However, what you write about Spinoza is deeply, factually wrong, and only displays your lack of any real knowledge on the subject. You are satisfied in your ignorance, and can’t be induced to cure it. Your loss, but it reflects badly on you.

I, on the other hand, have profound appreciation for Newton and Darwin, but rank Spinoza higher.

About all those digits – and an unproved conjecture, in case anyone is looking for something to do:

“The digits of π have no apparent pattern and have passed tests for statistical randomness, including tests for normality; […] The conjecture that π is normal has not been proven or disproven.[21]”

https://en.m.wikipedia.org/wiki/Normal_number

And of course:

https://en.m.wikipedia.org/wiki/Pi

… about the Bible thing – I’m not going to make a big deal about how the Bible got pi wrong – some reading suggests everyone was close but not accurate for a long time. But if a bible thumper makes a big deal about how they were close, then we can say join the club – everyone was close.

More interesting facts:

pi is an eigenvalue.

pi is not a physical constant <– apologies I think someone said this here already

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