Pi Is Not Irrational

Proposal: Irrational numbers are not necessary to describe real-world empirical matters, like "How much concrete do I need to pour this traffic circle?" Let's explore this idea and see where it leads.

There are a wide range of calculated quantities we might call "practical pi", or "pi-in-hand". All of these are rational. One could make similar observations about "practical root 2", "practical e", and so on.

Does the fact that we don't use irrational numbers in real-world empirical measurements mean that we can never use any results from RealAnalysis??

NonStandardAnalysis may be somewhat adaptable.

NonStandardAnalysis is working with systems that actually contain infinite quantities, as opposed to StandardAnalysis which is approaching them via epsilon-delta limits. There's no way it could be used under the above program.''
Nelson's adaptation seems relevant here, but I have to concede that analysis, standard or not, would require extensive reformulation to work without infinites.

This point is very important. If you throw out analysis, you lose the best (in the sense of the greatest utility) mathematical tools yet seen for science and engineering. You are talking about throwing out the calculus, differential equations, probability theory, etc. Now much of what I have mentioned can be done in other ways (although they certainly have not all been demonstrated rigorously). It is not clear the extent of things you will not be able to reformulate, but it is clear that this programme would be a non-trivial effort. If you want to convince people that a move like this is worth considering, it seems to me that the first step is to present an argument that the potential gains are a least larger than these losses. I don't see anything resembling that argument here...

I didn't suggest throwing out analysis. I suggested the possibility of reformulating it. But before we do that, we have some much more fundamental questions to ask ourselves. Probably starting with a DefinitionOfRepresentation. Let us try to breathe before we try to crawl.

Counterargument

The statement: 'pi is not irrational' is false. pi is a real number, well defined within an axiomatic system. If what you really mean is 'pi is not physical', then that is something else entirely.

The proposal makes two logical errors:

The mistaken belief that analysis requires any physical support.
The mistaken belief that a model of the world *is* the world.

I *think* the original author was trying to make a case for the idea that a) the models we use in the sciences and engineering are the cause of some problems, and b) that a system that is not reliant on the field of analysis as we know it would improve things.

However, the original since author has not demonstrated a), and has not provided b), it is difficult to know exactly what the point of this page is.

I would certainly like to see some clarification of the point, preferably without irrelevant statements about fairies and orcs. If OP is unable to resist uch inanities I suppose the chances of something useful coming out of this page are small.

Not proposed

Q. To explore this idea and see where it leads, must we dispense with the very concept of irrational numbers?

A. No.

Q. To explore this idea and see where it leads, must we dispense with the very concept of Turing machines?

A. No.

Also, no proofs by induction.

Why no proofs by induction? Does a general case imply an infinity?

Doesn't there have to be the possibility of continuing the n to n+1 process indefinitely for us to be happy that the expression works for any n = m?

Does an inductive proof fail to apply to a finite set? If we were formal in our definition of a method of subdivision, couldn't we apply induction to this definition?

Yes it does, because the last element of the finite set fails the n+1 test. Therefore, the whole inductive proof fails.

No. The last element was dealt with when "n+1" found it. Since it was the last element, further use of "n+1" is not required.

Redefine "inductive proof", please.

Poor engineering of computing architectures due to concept of "infinity"

One example would be...

The VonNeumannArchitecture, naturally - based entirely on Turing's silly idea (WhatDoesHaltingMean). It makes little sense for most of the logic gates in your computing device to simply sit and maintain a static procedural representation while your CPU chugs away synchronously. The EmergentComputing movement is finding plentiful evidence that natural computing processes are nothing like this - and that they're dramatically more powerful than VN procedures. We've spent fifty years software-engineering in a cul de sac, while mushrooms and bees demonstrate more computing power than our multi-trillion dollar follies. See LeibnizianDefinitionOfConsciousness for more on this.

OK. Thanks. Please now explain how, if the early hardware designers had not had access to the concept "infinity", they wouldn't have laid the foundations for our current problematical state of affairs. Alternatively, please explain how, given the VNA, not being able to use the concept of "infinity" subsequently would have stopped us from getting to where we are now, or even enabled us to have built machines that work the way these natural computing systems do.

Let's try for both. 1) If we'd had no infinity in describing the possible lengths of bitstrings on Turing's silly machine, we'd have had to come to grips with the basic question about just what is modelled by that tape, and just what fundamental mechanisms are at work inside that read/write head. We'd have had to ask questions about how natural computations work, and how we might model them. Instead, we've built the world's biggest and most expensive RubeGoldberg device.

2)The VNA without a concept of infinity requires investigation of the ontology of exclusively contractive address spaces. We have actually gone a little way down that route with FractalTheory? ... but no one except Wolfram has publicly suggested using Fractals for intentional computation. The VNA is a hideously overcomplex and inappropriate architecture for generating any kind of fractal, much less one with EmergentComputing abilities.

Poor engineering of communications architectures due to concept of "infinity" One example would be...

The idiotic distinction between processing and routing "data". All those towering procedural protocol stacks. We create address spaces without semantics, then centralize the processing of procedures to map semantics onto those spaces. There's no reason that network routing elements can't be treated as computing elements - and there are already Internet hacks that do exactly that by exploiting the natural structure of the net's switching fabric.

OK. Thanks. Please now explain why this idiocy is intimately connected with the very concept of infinities.

Let me come back to this one.

(I've had this thought too, the one not in italics. However, I eventually had to reject it. Non-Von Neumann algorithms are too "powerful"; we are incapable of predicting any but the simplest of them, and if anybody has actually harnessed chaos on a grand scale (you know, chaos-based web browser or something, not a mathematical toy) instead of frantically trying to merely reduce or predict the damage it does, that would be news to me. If you can't come up with something off the top of your head that isn't VN, but could have been implemented in 1960s technology, then maybe it's not so easy. "Powerful" gets scare quoted above because again, if anybody has ever chosen a significant problem in advance and harnessed a chaotic force to solve it, it would be news to me; chaos is more powerful then VN in theory but due to the near complete inability to control it, that power is effectively powerless. (Evolutionary computation is the closest it gets, and we're a long ways from evolving even the simplest UNIX utility.... yes, in our BraveNewWorld we wouldn't have UNIX utilities but if you can't get to that complexity level, you can't do anything else, either; complex problems exist.) I can't prove there's no big breakthrough waiting here, but until I see some actual evidence of it, instead of endless HandWaving of the NewKindOfScience type, I'll stick to this opinion.)

Aren't fractals a concept initially based on infinities, as well?

The mathematical objects we call fractals are the limits of infinite processes, and that would seem to stop us using them for anything, under this new scheme.

If you possess an IterativeFunctionSystem, what suggests you need to iterate it to infinity? Have you ever heard of anyone iterating it to infinity? Do you in your wildest imagination expect anyone to iterate it to infinity? If not, then perhaps you would like to ask yourself whether such fantastic limits are necessary to the utility of fractals.

You appear to have missed the point (again?). Fundamentally, IFS systems rely on Banachs contraction mapping principle, which is a result from analysis, and gives you conditions under which such a map is convergent. Of course nobody is expecting you to iterate an IFS infinitely. However, it is the results of analysis which allow you to say a) this map will converge, and b) give you a bound on the error at a particular iteration.

If you throw this away, you need to devise under your new system a way by which you can characterize the IFS (i.e. what maps converge and what maps don't), give a rate of convergence, and give a bound (even better if it is exact) on the error. Note that a fixed lattice representation will not necessarily be good enough for all sets you might wish to consider. Note also that in general (this is not so much an issue for fixed point iterative schemes like IFS) that merely measuring the current inter-step error is not good enough to describe convergence.

Do you have such a result?

They also rely on the density of the real line, complex plane, etc.

In the standard formulation, sure. Since we're talking - speculatively - about a reformulation according with empirical phenomena, we needn't quibble about the standard formulation.

Then again, the occurrences of fractal-like structures in nature only ever have a finite number of levels of self-similarity.

Yes.

If we could look at fractals based on finite approximations thereto, we could look at VNMs based on finite approximations thereto. The above author seems to think considering VNMs at all is a very bad thing, since there are all sorts of situations where they make things difficult and presumably none where they're a useful model.

I'm afraid that's not what the author means. VNMs are very useful indeed - until we have something better. What's more so much good engineering has been thrown at the VNM, we'd be silly to just dispense with it. The best idea is probably to enlarge the whole game of computer science and then see where all the current pieces best fit.

Presuming, naturally, that there is a practical way to enlarge that game. EmergentComputing seems a little undisciplined yet. All we've really noted on WhatDoesHaltingMean is that TuringMachines aren't necessary; for all we know they might still be sufficient.

Poor engineering of other sorts due to concept of "infinity" One example would be...

Good question. We'd have to look at specific design techniques - finite element analysis and multidimensional scaling - probably beyond the scope of this conversation and certainly beyond my qualifications to speak.

Also, it would be nice to see...

How to expand pi without infinities

We're told: Normalize your unit interval to the whole range of variation of all combinations of all relevant sensors, and consistently label subdivisions of this region through the use of fractions. When you do that, the result looks like...

Which part of these instructions was too hard to follow? If you don't iterate much, pi looks like 3. If that doesn't satisfy your intent, iterate a little more.

Well now we get down to it. Is this different from what I was talking about on ThereAreNoCircles? Seems not. Fair enough, no argument so far. What's missing for me is where this connects up with denying ourselves use of the real numbers for anything whatever.

There are no real numbers, circles, or similar boojums. They simply don't pertain to any natural phenomenon at all. Ergo anything you use 'em for is a fantasy. I have nothing against fantasy - hobbits, orcs and circles all seem like wonderful entertainments. But because ThereIsNoInfinity there can be no irrational Pi, and that's all this page set out to say. So we agree, down pens, that's that.

This is silly. Of course there are real numbers and circles, in the same sense that there are notes in a musical scale. These are not 'fantasy', they are called *abstractions*. Are you seriously arguing that abstraction is in general a bad thing? If you want to point out the simple fact that real numbers and circles are not *physical*, that is something entirely different, and hardly a point of contention. You could make an argument that particular abstractions have less utility than others, or that a particular abstraction has led to problems, but you haven't done that here.

: Of course, the same case can be made that integers do not exists, as there is nothing inherently physical about '1'. I have '2' apples, but they aren't equal, so by some definition I could say that I have 1.9999999937354364 or 2.00000000123514325 or whatever. Likewise, a 8.5x11 sheet of paper isn't really 8.5 by 11 inches, just very close to it. The same logic that denies the existence of an irrational 'pi' can be brought to bear against the existence of integers. This is why we deal with uncertainties for accurate measurements.
: The 'pi' as irrational view is useful because of its relevance to reality (which is exactly as much relevance as integers): I know that if I tie a string down, and tie a piece of chalk to the other end, it'll make something resembling a circle. As I increase various constraints, the ratio to the size of that string to the length of line drawn approaches a constant value. We call that imaginary value pi. And we use it because it has proven to be a remarkably accurate method of determining various things about various other things which resemble circles, waves and other such 'imaginary' thing. Of course, they are no more imaginary than the 3 'apartments' (another imaginary construct) on my block, the 5 'cars' parked outside my work, or the '120 000 kilometers' I've put on my 'car' with the '13 inch' 'wheels'. On the other hand, I've never seen anything which is arbitrarily similar (by which I mean as close as I care to look) to an orc, hobbit or boojum. Those simply aren't useful abstractions in reality.
: -- WilliamUnderwood

OK, I'll run with that. Perhaps we should BeRationalAboutNumericalAnalysis

With the exception of the discussion of infinities in computing theory, the subject of this page is a restricted form of mathematical intuitionism (See http://www.wikipedia.org/wiki/Mathematical_intuitionism). This arose in mathematical philosophy more than a century ago, and was formalized by Brouwer. Before we re-argue the matter, reading chapter 10 of Morris Kline's Mathematics: The Loss of Certainty ISBN 0195030850 would be a worthwhile endeavor.

Thanks for the pointer! Shall dig in.

Or, more succinctly: PiIsNotIrrational for nonexistent values of "not irrational". And, naturally, for nonexistent values of irrational too.

The name of this page should really be PiDoesNotExist? or perhaps RealNumbersAreNotReal? ...

Maybe I'm being a stick in the mud, but why the pseudo-intellectual, pseudo-scientific claptrap presented above is even being debated in this forum is beyond me. (It does, I suppose, have amusement value). We may as well discuss why TheEarthIsFlat and TheSunOrbitsTheEarth? while we are at it, along with the medicinal properties of leeches, and alchemists' formulas for converting base metal into gold.

Once we got over the idea that leeches were used exclusively by quacks we were free to observe that they have a great deal of medicinal value.

So are you suggesting that we should spend serious time debating the consequences of the Earth actually being flat?

What I don't understand about this page is what can possibly be gained from throwing away the levels of abstraction provided by proper use of infinities. For hundreds of years, mathematicians have built upon the power of abstraction to develop these tools - why do you want to discard them?

Pi exists in Mathematics

The misguided discussion referenced above confuses the world of Mathematics with the "RealWorld". Of course, circles and points and pi do not exist in the RealWorld. They are, however, very useful in modeling some real world problems so as to allow us to use mathematics to solve them.

Circles in the real world are not the same as circles in Mathematics. A real world circle would probably be closer to a Mathematics cylinder, but even there the roundness is not perfect, and the sides are neither parallel nor perpendicular to the base. There is no absolute center from which to measure a radius. There is no perfect plane slicing through the cylinder to use to measure the circumference. The measure instruments are not perfect lines. The measure itself is only reproducible to a certain degree of accuracy.

There are two similar, but different things under discussion. 2*pi*r is an acceptable answer in Mathematics, but an irrational measurement is a meaningless term in the real world.

The biggest problem I see here is that we are conflating the 'property' and 'thing'. Infinity is a 'property' of a process in our universe, it signifies something not unlike conservation of momentum. In other words, if nothing causes a process to terminate, it will continue. 'Infinity' is merely a symbol we use to describe an observable property; sometimes we also use the word 'divergence'. In any case, no one claims that 'infinity' is an actual object.

Then apparently you don't understand the Block Universe of GeneralRelativity.

Pi versus Universal Constants

Somebody compared the arbitrary-seeming universal constants (speed of light, etc.) to Pi. However, Pi is mathematically derived, not based on empirical measurements (although it matches empirical measurements). Thus, they are not the same kinds of thing. Another universe will not have a different value of Pi, at least not without a completely different kind of geometry being the base model.