Apparently, most mathematicians believe in the existence of infinity. But there are a few practitioners of finitism and ultrafinitism.
What are the consequences of these different approaches and do they affect calculus or pre-calculus? They don’t, right? Because all the proofs I can recall from my introductory calculus classes used limits: even if infinity doesn’t exist, you can still approach it with an arbitrarily large number for your application (which is still far short of infinity).
This topic arose from a number of tangents to this highly enjoyable thread:
Numbers don’t exist in the sense of being actual objects interacting within the spacetime continuum. For example, electrons exist. You’ll never find a five floating in space.
Numbers are abstractions used by humans. Very useful abstractions, like “red”, “unicorn”, and “antidisestablishmentarianism”. Many numbers can be directly used: “five disestablished red unicorns”. Others require more training to be useful: G_{\mu\nu} + \Lambda g_{\mu\nu} = \kappa T_{\mu\nu}.
Ultrafinitism would kill any argument with limits. If you cannot use any number >10^{12}, say, then you cannot use any \epsilon<10^{-12} and then limits become meaningless.
As for finitism, I think limits become possible, but what you probably cannot do is prove much about them. For some insight into this, you should look at Errett Bishop’s book, Foundations of Constructive Analysis. Bishop is not a finitist and he uses a restricted form of induction and proves things like that polynomial functions are continuous. One curious fact (not a theorem): every constructible function is continuous. It’s not a theorem because it requires non-constructive methods to prove. But Bishop says that he doesn’t think one could construct a discontinuous function (using his definition of construct).
Although Bishop is not a finitist, his book will give some insight into what a finitist might do (not much, to be sure).
As far as I am concerned ultra-finitists might as well be flat-earthers, for all the good can come out of them.
that attempt to establish that, as far as physics goes, finitistic arguments are sufficient. However, I suppose @Pleonast must be correct that this does not reflect how physicists and mathematicians work— they just use classical mathematics.
Mathematicians, for the most part, use classical mathematics. Physicists, however, in actual work, tend to use bastard mathematics, on the grounds that they work. And every so often, a mathematician will take some particular piece of physicist bastardy, and come up with some way to legitimatize it.
I take it that there’s some sense in which \frac{1}{1+e^{1/x}} is nonconstructible?
Plato had a lot to say about what it means for numbers or mathematics to exist. I’m not convinced we have progressed since then. Which makes this recent cartoon rather appropriate.
In any interval that avoids 0, it is certainly constructible. I guess that you mean there is no constructible function in the interval [-1,1] that vanishes at 0 and is otherwise \frac1{1+e^{1/x}}. Truly, I don’t know. I suspect not, but such a function would be continuous at 0.
Physicists (and most mathematicians) were perfectly happy to use infinitesimals for discovering things. But the mathematicians felt the need to actually find \epsilon,\delta type arguments for publishing proofs. Until
Abraham Robinson came along around 1960 to find a very clever way to use the Axiom of Choice to actually construct non-standard numbers that included infinitesimals and rebuild calculus along those lines. He wrote an elementary calculus text exploiting this (although the actual construction had to be relegated to an appendix), but it never caught on.
As Francis_Vaughan noted, this touches upon some very old and very well traveled parts of philosophy and is by no means a settled issue (though frankly I think your framework is workable in a practical sense).
The color “Red” is a concept (corresponding to light emissions at a particular range of frequencies). The interior of a watermelon is red, and it’s red regardless of whether humans are around to observe it (though perhaps you could make an argument that mammalian eyes would be useful assumptions). “Watermelons typically weigh more than cherries, on average”, uses the concept of weight, something that also has an existence outside of human experience.
“Leaflets three, let it be” aka “Poison Ivy typically has three leaflets”, is also true independent of a human observer. So numbers exist, just as mass exists, just as any number of characteristics exist, though you won’t see mass or color floating in the air any more than the concepts of three or four.
It’s interesting that truths can be ascertained without empirical observation, yet that is what mathematicians do. As I understand it, all these truths are conditional since they rely on axioms.
@Hari_Seldon (or anyone): What’s the easiest textbook on real analysis?
I confess I don’t grasp Hari’s argument. I mean if you can do this:
lim x → infinity,
you can do this:
lim x → Really big number, 3+ orders of magnitude bigger than the problem you’re working on. (3 orders of magnitude is nothing, but honestly that may be all you need). 300+ orders of magnitude - an arbitrarily large number short of infinity, but which infinity would approximate in your application.
That said, following my discussion above I don’t see any clear reason to rule out the existence of infinity any more than I would rule out negative numbers based upon the fact that no trees contain negative numbers of fruit, no houses negative numbers of bricks, no organisms negative numbers of cells. Some mathematical concepts simply have more general applications than others. Infinity may or may not have any physical counterparts (as the universe may or may not be of infinite size) but regardless the concept could exist even if applications of it are only indirect.
You cannot do limits in any useful sense without arbitrarily small positive numbers whose inverses will be arbitrarily large numbers that ultrafinitists don’t accept. This does not apply to ordinary finitists who do allow arbitrarily large–and therefore arbitrarily small positive numbers.
Really, ultrafinitism opens up all sorts of cans of worms. Like, suppose that we call the largest number N. Then N*(2*0.25) is a valid number, but (N*2)*.25 isn’t. And therefore, the Associative Law doesn’t work any more. It’s tough to do math without the Associative Law.
I’m imagining a math where the laws break down as you travel past numbers that humans dare not explore. Somewhere around warp 12.
Speaking as an outsider, it’s somewhat interesting that at the very least it’s not straightforward to construct calculus from the ground up on ultra-finitist/ ultra-intuitionist/ actualist/ concretely representable/ visualizable foundations. Or so I gather from A.S.Troelstra’s History of constructivism in the 20th century, obtained from a wiki footnote. The author goes further:
The first author to defend an actualist programme, was A.S. Esenin-Vol’pin in 1957. He intended to give a consistency proof for ZF using only ‘ultra-intuitionist’ means. Up till now the development of ‘actualist’ mathematics has not made much progress — there appear to be inherent difficulties associated with an actualist programme.
In practice, I expect that if an ultra-finitist needed calculus, they’d just use small finite numbers in place of infinitesimals. Which is, after all, what real scientists and engineers often do.
Though scientists and engineers still have the luxury of decreasing the size of their “infinitesimals”, if needed. With fixed lower and upper bounds, you end up with something more like video-game design. Most quantities in video games are actually integers, just with small enough units that they look continuous (even if the units displayed to the player are much larger), because that makes the code more efficient. But there’s a bit of an art to setting the scale of units for each sort of quantity, to ensure that all relevant ratios of those units also work out well.
The Troelstra citation suggests that the ultra-finitist research agenda ran into difficulties starting in the mid 1950s.
Positions need to be discrete as well. If positions are continuous, then the list of possible positions in a finite universe is infinite. Similarly for other metrics.
Speaking as a non-physicist, neither the continuous nor the discrete model of distance is especially intuitive, not that my personal anthropocentric intuition matters at those scales.
Basically, only by their proponents, but not by any other mathematicians. Bishop’s constructionism is taken somewhat seriously. At least he produced something you could call mathematics. His book is actually quite interesting. One thing is that certain paradoxical constructions become impossible. On the other hand, certain exremely reasonable principles also disappear. Like the existence of a basis for any vector space, especially infinite dimensional ones.
But if positions are quantized at, say, the Planck length, then we are back to a finite number.
However, I don’t think either of these things work as a limit to numbers. First, just because the observable universe is finite doesn’t mean that the whole universe is. Second, if we took the number of possible positions in the observable universe as a limit, we could always measure something even bigger - like all possible arrangements for all particles in the universe in a grid of Planck sized positions.
If positions are finite on a planck grid, what happens if you move one planck length south, one planck length west, then one planck length northeast?
How far away are you from your starting point?
