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This may not qualify as a philosophy question exactly, but I would argue that potential applications of pure mathematics are in the bounds of philosophical interest.

Many innovations in pure mathematics, such as Riemannian geometry, turn out to have totally unanticipated applications in physics. Is this true of Cantor's work? Have his transfinite numbers had any practical applications in physics, technology, or engineering?

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    The idea behind ordinal and cardinal constructions, namely stacking completed recursions on top of each other in a radically unrestricted manner, is a cornerstone of modern theory of computational and logical complexity, see e.g. lightface Borel hierarchy en.wikipedia.org/wiki/Borel_hierarchy Logical constructions like ultrafilters of non-standard analysis also rely on it. This is not straightforwardly applied as Riemannian geometry, but the latter was explicitly developed from Kant inspired musings over perception of physical space, so Einstein's use of it is not entirely surprising. – Conifold Sep 29 '15 at 2:49
  • Thanks. Much of this above my head, but good leads for browsing. Never heard that Kant inspired Riemann... more often said that Gauss, Riemann, et a al "disproved" Kant. Though I have also heard that Kant was well aware of nonEuclidean theories and felt they had no impact on his foundational role for "space". – Nelson Alexander Sep 29 '15 at 13:11
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    Herbart, who succeeded Kant at Königsberg, explained Kant's philosophy of space in a way that made it popular among 19th century scientists, including Helmholtz, Grassman, Riemann and Poincare. Their idea was that our perception isn't as specific as Kant thought and only singles out locally Euclidean geometries, the physical one of them to be determined empirically. See HSM thread hsm.stackexchange.com/questions/657/… – Conifold Sep 29 '15 at 20:05
  • There seem to be applications of transfinite sets in high-energy physics. Try doing a Google search on "transfinite high-energy physics" and check out some of the results. I stumbled on these by accident a month or so ago and have not researched these any in-depth. For myself, I am undecided whether these are implications for "actual" infinities in some sense of "reality", or merely mathematical techniques used in building various "models" as an aid to understanding various conundrums in physics. Hence the "practical" implications may also be up in the air. – Jeffery C. Niemuth Nov 23 '15 at 6:32
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    @Mozibur. I have read this but don't recall the details. I believe there was actually some moves towards non-Euclidean geometry before Gauss. – Nelson Alexander Nov 23 '15 at 21:33
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Not immediately. Transfinite sets have affected logic and other areas of pure mathematics, but as far as I know, they have had no practical application. Within ZF(C/U), you can fit all of the rest (ie non-transfinite) mathematics in the first two/three infinite levels.

  • Well, by definition all of non-transfinite mathematics needs only the first "level" of infinity. – Noldorin Sep 28 '15 at 17:30
  • Fair point. I meant that all the other areas of mathematics - some of which have big practical applications - doesn't require anything more than the second infinite level. – Carlo Lori Sep 28 '15 at 17:32
  • Indeed. And you can even get around the requirement of $\Aleph_1$ sets (e.g. the reals or complex numbers) by reformulating analysis. This is often done in constructive settings, and it has been shown to be quite sufficient for the great majority of analysis. :) – Noldorin Sep 28 '15 at 17:39
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    @Noldorin The cardinality of the reals is not known. It could be Aleph-1, but no-one knows. The axioms are not strong enough to determine the cardinality. – Nick Sep 28 '15 at 20:18
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    @Noldorin Harvard held a recent workshop on CH, logic.harvard.edu/efi.php#multimedia The majority view seems to be Cohen's pluralism, CH is indeterminate, Woodin's view is that it fails, and even Godel's informal arguments make continuum into aleph 2, not aleph 1. MSE thread on CH does not favor it either mathoverflow.net/questions/23829/… What are the reasons to believe in CH unless one subscribes to the constructible universe, which is seen as an aberration by most? – Conifold Sep 29 '15 at 2:37
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The short answer is no, transfinite sets are not in and of themselves practical. They're certainly not "concrete" (this can be formalised by saying they're not computable).

The long answer is, there are many different approaches to the foundations of mathematics and even set theory, and ZF(C) and other transfinite theories are only a few (albeit the most dominant, for historical reasons). These transfinite theories are very powerful, and can be used as settings for the entirety of mathematical knowledge. However, there are also classical and strict finitist theories, that themselves come in various philosophical forms, and are considered sufficient to express the great majority of mathematical thought, including virtually all of 19th century arithmetic, geometry, and analysis, and much 20th century maths too. There are other foundations still that take computability as the fundamental concept, and these necessarily cannot be transfinite.

It's worth saying however that research (and even less usage) of non-ZF systems is still relatively low-key in the mathematical world. Although most working mathematicians don't even concern themselves explicitly with such low-level things as the axioms of ZF, many do occasionally resort to them, and there are set theorists who effectively study them for a living. Set theory (implicitly meaning ZF set theory) is a relatively niche area within mathematics however, and modern results from it are not generally given much attention within the wider mathematical community.

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Much cryptography relies on number theory, and much number theory relies on the existence of large cardinals.

For example, the Weil conjectures (at least for curves, and presumably more generally via the same sorts of constructions) have clear immediate implications for cryptography, and the proof of those conjectures relies on the machinery of Grothendieck universes, which is to say that it relies on large cardinals.

  • Not sure why the down vote, but I voted back up. I don't know the details or how direct the application might be, but it does seem that cryptography might qualify. – Nelson Alexander Sep 29 '15 at 13:02
  • I like this answer too. I'm surprised I had forgotten this example. Certain features of our number theory can be explained as the "imprint" of large cardinals. – Nick Sep 29 '15 at 16:11
  • @nick r: I'd heard, or rather read the same in some news article; it was good to see some practical application of large cardinals, in the sense that mathematicians think practicality... – Mozibur Ullah Oct 3 '15 at 11:12
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    Weil conjectures for curves, which are 'used' in the applications of number theory, were proved by Weil without use of large cardinals or the later developments of Grothendiek, but really Hasse's theorems from before are already enough. – Dror Speiser Jan 10 '16 at 8:59
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    Most number theory does NOT depend on large cardinals. Can you source the Weil conjectures have implications for cryptography (not saying it's false, just requesting a source). As @DrorSpeiser notes, the Weil conjectures do NOT depend on Grothendieck universes, although they do depend on other work Grothendieck did. – barrycarter Mar 17 '16 at 3:03
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Here are a few applications of infinite ordinals that I believe almost certainly count as practical.

Well-orders on various countable sets which are order-isomorphic to infinite ordinals are very important in computational mathematics.

The theory of Grobner Bases in computational algebraic geometry requires one to fix an ordering on the set of monomials in the polynomial ring in n variables over a field. One such ordering is the lexicographic ordering, which is of order-type n*omega, where omega is the first infinite ordinal. Real-world applications are found anywhere where solutions to systems of polynomial equations are needed.

I'd also like to argue that transfinite induction used in proofs such as Gentzen's consistency proof is in some sense finitary. It establishes that if the result in question is inconsistsent, then in finitely many steps one could produce a contradiction, since any descending sequence of ordinals is finite.

Another application that no one mentioned here is the real numbers. These form a set which is obviously infinite and in fact is uncountably infinite. People might object that in practise one never needs an actual infinity of real numbers for any particular calculation, but any finite formulation of real numbers ends up being prohibitively convoluted. That the theory of infinite sets gives us an elegant and simple framework for reasoning about the reals is a practical application in itself.

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There are two theories of the infinite, the ordinals and the cardinals; neither have a direct application to the world in a concrete manner; as a generalisation of number, they have no referant as say the number two can refer to two plums.

Similarly they don't have direct practical applications in physics or engineering.

However they do have utility within the body of mathematics for example Gentzen proved the consistency of Peano Arithmetic by allowing induction to a larger ordinal.

But here practical means outside of its own domain - we're not using ordinals to solve some problem of ordinals - but an important question in some other area of mathematical logic; this sense of practical is obviously different from the sense above; and also in the sense the question uses.

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    Gentzen's transfinite consistency proof is usually dismissed on the grounds precisely because it's transfinite. – Noldorin Sep 28 '15 at 21:01
  • @noldorin: it depends, I would say, on your philosophy of mathematics; do those that dismiss this proof also dismiss the higher ordinals? – Mozibur Ullah Sep 28 '15 at 21:06
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    Even Hilbert, an ardent supporter of classical mathematics, ZF, and transfinitism, thought that infinitary consistency proofs were inadmissible. In formalism, they're accepted if you're working within ZF, but consistency proofs must be necessarily done outside of ZF, and one should be able to verify them by the intuition, according to Hilbert and most prominent classical mathematicians. Constructivists of course are even more adamant against their validity! – Noldorin Sep 29 '15 at 22:35
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I largely agree with the other answers posted, that, in a "real-science" context, there is no apparent practical applications of transfinite numbers or their arithmetic. But I'm inclined to want to come up with something, so you may consider this - and admittedly this is stretching the meaning of practical to breaking point :

In cosmology, the popular eternal inflation model makes it possible that actual infinities exist in our universe. For example, the universe could be infinite in extent, or there could be infinitely many universes in the multi-verse, or there could be a temporal infinity, etc..

If this is the case, then the question of assigning cardinals to these infinities would be one such application.

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    This certainly sounds like it would have practical implications for the travel industry. – Nelson Alexander Sep 28 '15 at 23:31
  • @NelsonAlexander I spoke too soon. In fact, David Deutsch has used the cardinality of the continuum in the formulation of his multiverse theory. According to Deutsch : “within each universe all observable quantities are discrete, but the multiverse as a whole is a continuum." Thus, there is an uncountably infinite cardinality instantiated in our multiverse - at least according to Deutsch. See en.wikipedia.org/wiki/Digital_physics . So transfinite numbers have made an appearance in our physics. – Nick Sep 29 '15 at 15:59
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Yes and no.

Yes, in that the basic notion of cardinality is fairly important, in particular whether a set is countable or uncountable. If it's countable, you can parameterize all the elements as arbitrary-length finite strings on some finite alphabet. If it isn't, then you can't, and you need to either look at infinite strings, successively better finitistic approximations, or truncated infinite strings ending in "dot dot dot." This basic question arises commonly enough in most fields of both pure and applied mathematics that it's something you ought to know.

No, in that beyond the above fairly basic question of countability, transfinite quantities don't seem to get involved all that often. Almost all of applied math can be formalized using only three infinite cardinals - that of the natural numbers, that of the reals, and every so often, that of the power set of the reals. More to the point, most of the most powerful applied math techniques we have involve finitistic approximations of problems, such as in numeric computing, so that we are only computing things to arbitrary accuracy anyway and infinity doesn't really get involved in any meaningful sense.

The essence of the problem, unfortunately, lies in that the theory of transfinite cardinals and ordinals hasn't quite "developed" in a concrete way, similar to the rest of math, as Cantor hoped it would when he first created it. We cannot prove even basic things like if the reals are the next-largest cardinal after the naturals (the continuum hypothesis), which we now know, thanks to the results from Cohen's celebrated landmark paper on "forcing," is independent of ZFC set theory. In general, the structure of the cardinals is heavily dependent on the set theory, so to some extent you can "choose your own adventure" - yet there is no consensus on how to choose or even if there is a "right" way to choose.

As a result, it's difficult to use transfinite numbers as a core technique in applied math when much of the very basic questions are still open and buried in murky set-theoretic foundational mess.

The theory of ordinal numbers, on the other hand, does not seem to run into quite as much set-theoretic trouble right out the gate. I haven't seen them used that much in applied math (although certainly on occasion), probably because of the emphasis always being on numeric computation, which typically involves finitistic approximations of everything anyway.

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There is a clear no.

Transfinite sets cannot have practical applications because of two reasons: First, there is nothing transfinite in the universe, at least in that part accessible to us. And second, transfinite set theory is a self-contradictory system, not better and not more useful than astrology.

There is a society of believers in this system who will downvote this answer and who have upvoted provably wrong answers like number theory relies on the existence of large cardinals or cryptography relies on the machinery of Grothendieck universes or Gentzen "proof" by induction to a larger ordinal or the set of real numbers which cannot be distinguished and used but have a well-order.

To prove my case for the unbiased and objective reader here is one of many blatant contradictions between mathematics applicable to real science and transfinite set theory:

Scrooge McDuck receives 10 $ every day and spends 1 $ every day, always the oldest one of his possession.

According to transfinite set theory he will go bankrupt in the infinite limit. This result is crucial for set theory. Without it, the fundamental distinction between countable and uncoutable sets would break down.

The argument is simple: For every dollar we can determine the day when it will be spent. The argument is wrong because: For every dollar we know that it is followed by at least 9 other dollars. This is overlooked intentionally because this fact prohibits the exhaustability of infinite sets - a basic requirement of set theory.

Of course in mathematical analysis there is only an improper limit: Scrooge McDuck will get infinitely rich. He will never become bankrupt, not even in any limit.

Every mathematical statement or theorm that relies on transfinite set theory is wrong or at most true by chance.

  • Coming up with a bijection does not mean Scrooge will be bankrupt. I think you're missing the point of a transfinite ordinal "number". It's just supposed to be a canonical representation of an ordering. If you're annoyed because it allows for an ordering like {1,2,3,...,0}, that just means, if you take two numbers m,n, from N, the ordering above means m>n in just the same way as normal, but if one of m,n is 0, then it is greater than the other. You can say you don't like that generalization of ordering, or the generalization of " number", but you can't really call it false (or true). – Franz Aug 18 '17 at 19:34
  • My argument has nothing to do with ordering. It is simply this: Unless the set of not spent dollars is empty in the limit and unless the set of not enumerated rational numbers is empty in the limit, set theory fails to prove the existence of a bijection with |N. – Heinrich Aug 18 '17 at 20:58
  • So you mean you don't accept the validity of a bijection from N to a subset of N? E.g. you don't accept f: N→N: 2n? Or the identity function on N? Because that's all the bijection is in your example. If each dollar is labelled as it comes in, you can work out exactly which day each dollar is spent. Of course for any day n, he will have 9n in his balance. The bijection is nothing more than the mapping between the two sets. It seems you have more of an issue with the idea of infinite bijections than the idea of ordinals specifically, would that be accurate to day? – Franz Aug 18 '17 at 22:27
  • I do not accept that the infinite sequence of |N has an end or can be exhausted. But that is required for Cantor's arguments. All rational numbers must be counted to prove equinumerosity. The "Cantor-list" must contain the complete set of all natural numbers to prevent insertion of the diagonal number. That idea is simply wrong. There is some confusion between properties that can be stated for all natural numbers (like: all natural numbers are integers) and others which cannot (like McDuck can issue all his dollars). The whole edifice of transfinite set theory is built upon this mistake. – Heinrich Aug 19 '17 at 11:48
  • Ie you don't believe in sets that are not equivalent to subsets of the N? That's fine, but there's a lot of maths you're missing out on if you don't believe in R. And I'm not sure you understand the meaning of bijection, because scrooge will pay each of his dollars. Each dollar he points to, you can tell him the day he will spend it. That's all the bijection means. It doesn't mean he will go bankrupt. That's not what bijection means. – Franz Aug 20 '17 at 8:40

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