# Is mathematics infinitely regressive?

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.)

• Related: this (possible duplicate?) and this Dec 1, 2012 at 3:16
• I'd go with the first - mathematics is full of paradox, circular reasoning and infinite regress, as well as stretches of air-tight logic. Dec 1, 2012 at 3:23
• @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. Dec 1, 2012 at 3:27
• 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 1, 2012 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... Dec 1, 2012 at 17:12

# 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.

• 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? Dec 5, 2012 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)!).
– DBK
Dec 5, 2012 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. Dec 6, 2012 at 0:14
• @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 6, 2012 at 0:51
• @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. Dec 6, 2012 at 1:39