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Agrippas trilemma states that formal systems are either self-circular, infinitely regressive or axiomatic.

Its commonly taken that mathematics is axiomatic. However just as mathematicians can build ever more elaborate structures, they can painstakingly dig-down into foundations.

Does this mean that mathematics is actually infinitely regressive, it's just that the diggig is a bit slow?

After ymars comment, I emphasise I mean formalistically.

(From another perspective mathematics is what mathematicians trained at certain schools do; similarly to some contemporary description of art by critics, art is something that artists trained at certain schools do. But one could argue here, the critics have just thrown in the towel.)

DBK
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Mozibur Ullah
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    Related: [this](http://philosophy.stackexchange.com/questions/1848/is-mathematics-founded-on-beliefs-and-assumptions) (possible duplicate?) and [this](http://philosophy.stackexchange.com/questions/3716/is-mathematics-always-correct) – commando Dec 01 '12 at 03:16
  • I'd go with the first - mathematics is full of paradox, circular reasoning and infinite regress, as well as stretches of air-tight logic. – Mozibur Ullah Dec 01 '12 at 03:23
  • @commando:I guess what I'm really questioning is the tripartite structure of the trilemma. That an axiomatic foundation can always be critiqued, so we must end up in an infinite regress, or a circular argument. – Mozibur Ullah Dec 01 '12 at 03:27
  • Mathematics isn't infinitely regressive because mathematics is what mathematicians do. And mathematicians, generally, do not "painstaking dig-down into foundations". Eventually it is all built on a gentlemen and gentlewomen's agreement. – ymar Dec 01 '12 at 13:31
  • @Ymk: fair enough, but there is a foundationalist movement within mathematics, we have ZFC, the peano axioms, material set theory, homotopy type theory, category theory advancing the notion of what foundations mean. Any rigorous work in any direction is 'painstaking'. Actually your comment reminds of what some contemporary critics say about art, it is what artists do. Critics in previous eras may beg to disagree. I wonder what Duchamp would say about it... – Mozibur Ullah Dec 01 '12 at 17:12
  • @Ymk: I'm not saying all mathematicians do, but a few have & still do. – Mozibur Ullah Dec 04 '12 at 21:51
  • Your [addition](http://philosophy.stackexchange.com/revisions/4341/3) doesn't really add to the question. Could you rephrase it or perhaps remove it? – DBK Dec 05 '12 at 13:42
  • @DBK: Ok, I'll rephrase it. – Mozibur Ullah Dec 05 '12 at 16:08

1 Answers1

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Mathematics is not "infinitely regressive"

Its commonly taken that mathematics is axiomatic. However just as mathematicians can build ever more elaborate structures, they can painstakingly dig-down into foundations.

Does this mean that mathematics is actually infinitely regressive, it's just that the diggig is a bit slow?

Mathematics is not infinitely regressive, because the establishment of a set of axioms from which all interesting theorems follow has been in fact not only finite, but realizable. Specifically a set of axioms cannot be further analyzed if all the axioms are independent.

An axiom P is independent if there are no other axioms Q such that Q implies P.

The analysis of axiom independence has been a very important search in establishing axiomatic systems and today's axiomatic systems are independent in this sense.

In case you wonder: The fact that it is possible to find different "foundations" for previously established mathematical results (i.e multiple groups of axioms for a given set of theorems) is a matter concerning the pluralism of foundational systems, not their being infinitely regressive.

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DBK
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  • Supposing there is a pluralism of foundations, ie a *multiverse* of set theories. Is there no possibility in the future that some *principle* could be found that then originates this *multiverse*? Do you rule that out? – Mozibur Ullah Dec 05 '12 at 16:54
  • @MoziburUllah: A "principle" from which ZF, ZFC, NF, NBG, KM, ***etc.*** would eventually "originate"? I would very much doubt it. And how would you find a common "principle" from which both ZFC *and* ZF¬C "originate"? So, yes, I would rule it out. The point here is that foundational systems are not just used by reconstructive means, but are battlefields to decide which interesting mathematical fact should actually be true (think of [CH (and ¬CH)](http://en.wikipedia.org/wiki/Continuum_hypothesis)!). – DBK Dec 05 '12 at 23:03
  • @DBK: in principle, ZFC and ZF¬C could arise as examples of two "set theories", in the same way that you can have two different commutative rings with additive identity denoted 0 and unity denoted 1, in which 1+1=0 and 1+1≠0 respectively. A common foundation could motivate ZFC and ZF¬C as objects of interest in some deeper theory in which the Axiom of Choice would not necessarily make sense as a basic postulate. On the other hand, we are not *required* to contemplate whether or not a "deeper potential foundation" exists, nor would it necessarily be unique anyway. – Niel de Beaudrap Dec 06 '12 at 00:14
  • @NieldeBeaudrap: I agree with you. Nice analogy wrt ZF and AC - I thought of the more classical example of euclidean and hyperbolic geometry in their historical formulations (i.e. devising respectively the parallel postulate and its negation) as "originating" from a "deeper foundation", namely Riemannian geometry. If that is what the OP is asking, then I would not rule it out. But it goes without saying that I also agree with your last point, so I don't see *this* as a way around the multitude of possible foundations. – DBK Dec 06 '12 at 00:51
  • @DBK Is there any proof that mathematics is not infinitely regressive? Linking Wikipedia entry which states "Proving independence is often very difficult" unfortunately does not suffice. – Sniper Clown Dec 06 '12 at 01:39
  • The "difficulty" alluded to on WP is about the more [general task](http://en.wikipedia.org/wiki/Independence_(mathematical_logic)) of independence proofs, esp. involving a rather advanced technique called [forcing](http://en.wikipedia.org/wiki/Forcing_(mathematics)). The bulk of work in this field is devoted to show the independence of 1) additional axioms expanding ZFC, and 2) mathematical claims which are *undecidable* in ZFC, see e.g. [What are some reasonable-sounding statements that are independent of ZFC?](http://bit.ly/7Fooeb) – DBK Dec 06 '12 at 02:10
  • @DBK: I would have made the same point as Niel, but he got there first. Another example might be the ['remarkable observation about large cardinal axioms is that they appear to occur in strict linear order by consistency strength.'](http://en.wikipedia.org/wiki/Large_cardinal). There can be no theorem to explain this, as there is no acceptable definition of what a large cardinal actually is. I never say in my question that foundations have to be unique. – Mozibur Ullah Dec 06 '12 at 02:22