Is Norman Megill's view of Gödel's incompleteness theorem compatible with what philosophers have said about it?

Question

Here is one recent and seemingly expert appreciation on the consequences of Gödel’s incompleteness theorem for mathematics:

Gödel’s incompleteness theorem showed that it is impossible to achieve absolute rigor in standard mathematics because we can never prove that mathematics is consistent (free from contradictions). (...) However, real mathematical systems have very few axioms, and these can be carefully studied. — Norman Megill, Metamath, A Computer Language for Mathematical Proofs (2019)

Metamath is a repository of computer-verified theorems, an activity which arguably belongs to computer sciences, but I assume that contributors to metamath are essentially mathematicians, and also mostly not philosophers.

See https://en.wikipedia.org/wiki/Metamath for details.

Given this, is Norman Megill's view compatible with, or perhaps even typical of, what philosophers say about the consequences of Gödel's incompleteness theorem for mathematics.

Answer 1

Part of ths relies on interpretation.

Your quote uses the expression, "standard mathematics." Given that section 74 of Kleene's "Introduction to Metamathematics" explicitly describes a difference betwen "informal mathematics" and "formal systems," what is meant by "standard mathematics"?

A similar distinction is found in one of Tarski's papers where he differentiates between actual systems which ought to be studied individually and the commonalities he intends to address. This may be compared to Aristotle's distinction between theses and axioms needed by all.

Megill's statement is certainly acceptable. But, has the case been made for a reduction of mathematics to the linguistic analysis attached to signatures from universal algebra? The first incompleteness theorem is slightly wider than formal systems, as they had not been fully developed. It used formalisms related to "Principia Mathematica" to address a metamathematical approach suggested by Hilbert. But the question of the reduction is still basically correct.

When one admits that the word "mathematics" is inherently vague, it becomes easier to see how statements like the one you quote deserve a great deal of scrutiny.

Philosophers do not teach mathematics in the same way as mathematicians in mathematics departments.

@speakpigeon

These days, it is unlikely that any algebraist does not use category theory. Thanks to the work of William Lawvere (on the shoulders of Grothendieck, MacLane and Eilenberg) category theorists make claims about "what mathematics is" that are quite different from what first-order logicians claim.

In May, a simple question about Bourbaki re-ignited a recurring argument on the FOM mailing list. At one point the category theorist, Colin McLarty chimed in because of a historical statement about MacLane's book, "Categories for the Working Mathematician." His statement is in the link,

https://cs.nyu.edu/pipermail/fom/2022-May/023302.html

Harvey Friedman, a strongly committed set theorist, responded by declaring that category-theoretic had no intellectually significant basis,

https://cs.nyu.edu/pipermail/fom/2022-May/023298.html

Harvey Friedman collects his paycheck in the mathematics department of Ohio State University. While it is true that Colin McLarty is in a philosophy department, Saunders MacLane had been in the mathematics department of my alma mater, the University of Chicago. The teaching of category theory is not some arcane "non-standard" mathematics.

And disagreements like these are not uncommon.

One often hears that set theory is the "foundation" of mathematics. That is true, perhaps, for logicists. However, Skolem's criticism showing that Zermelo's set theory cannot be categorical moves "foundations" to the language signatures of universal algebra. Chang and Kiesler describe (first-order) model theory as "universal algebra + logic." With the development of stability theory, a more modern summary from Wilfred Hodges summarizes (first-order) model theory as "algebraic geometry + logic."

The common theme here, then, is the "algebraization" of mathematics. And, Goedel's theorems are easily adapted to this in relation to universal algebra. To address "completed infinities" Hilbert distinguished between syntactic methods based upon the "sensible impressions of symbols on a page" and what can be studied through formal axiomatics. When structural induction is used to delineate the well-formed formulas of a formal language, the singular notion of a "language" presupposes a completed infinity. The structural induction, however means that any spelling of a well-formed formula can be recursively verified. Goedel's theorems depend upon these inductive relations.

Extension of the first theorem to a more general context involves understanding that axiomatic statements can sometimes be recursively verified as axioms. Thiis is why Zermelo-Fraenkel set theory and Peano arithmetic can have infinitely many axioms given through schemas. Importantly (because forcing can be justified on this basis), only finitely many axioms are assumed within any given derivation.

Goedel's theorems are often portrayed as differentiating provability from truth. Proofs, after all can be seen as lists of formulas. Arithmetization of formulas can be extended to lists of formulas. So this is how the theorem may speak of a truth from the standard model which cannot be proven.

Yet, notice that this is all based upon a characterization of mathematics reducing mathematics to linguistic forms. Following Paul Halmos, any algebraic logician will concede the validity of Goedel's theorems when applied to syntactic forms.

Is mathematics "just words"? Mark Steiner wrote a book on the applicability of mathematics. If mathematics is "just words" then what are applications? While they will deny any tangible relationship, set theorists use the word "belief" to compare "truths" between models of set theory. In his book, "Retreat to Commitment," Bartley outlines how the kind of foundationalism found wth the first-order paradigm has undermined "science." If mathematics is "just words," then applied mathematics are "just beliefs."

Steiner asked, by way of a conclusion, if applications of mathematics did not suggest a form of Pythagoreanism.

Among what others call "non-standard" mathematics, you will find homotopy type theory which makes an appeal to authority against Martin-Lof type theory. When Andriej Bauer and Steve Awodey could not engage significant consideration for this "new foundation" on the FOM mailing list they made very public "resignations" from the list. During this time, I discovered that Martin-Lof type theory advocates "verificationalism," thereby sidestepping the kind of distinction found in Goedel's arguments which had been so disconcerting. More precisly, proofs verify. So, the issues involving truth associated with Goedel's theorems need not actually be mathematical in nature.

Given "truth values" (or subobject classifiers) one can study "logic" without actually concerning one's self with truth.

When any mathematically-inclined person discounts these debates as "philosophy" or "history," they are engaging in rhetoric meant to avoid the hard questions which must creep in from philosophy. Because of the "wow factors" attached to information technology, "standard mathematics" thirty years from now may be quite different from the "standard mathematics" I learned thirty years ago. I had been at the University of Illinois as an entering graduate student (never completed because of an illness) when they started teaching calculus using Wolfram's Mathematica. And, there are already courses being taught with the proof assistant Lean.

The development of proof assistants had been a stated goal of the Fields medalist Veodovsky in his promotion of homotopy type theory. Again, these developments are a part of standard mathematics being put into standard curricula.

I know what I know because I have studied a single mathematics problem for 37 years. Yes, I am an autodidact -- or crank if that is the manner of your thinking. But, I am on record in this forum as advising any student of mathematics to avoid "foundations." They should like what they like and believe what they believe without concern for "what mathematics is." Such advice is not typical of a crank. To the contrary, I have found the study of a difficult problem to be entirely humbling.

I will consider the time I have spent with this narrative as successful if it has led you to reconsider your assumption on the meaningfulness of the expression "standard mathematics." All I have found is a diversity of opinions which cannot be reconciled.

Answer 2

The very first axiomatic system is Euclidean geometry. With the advent of formal mathematical logic in the early 20th C, Tarski formalised a substantial fraction into first order logic with a single primitive notion, point, and without any set theory. In some ways it went beyond what Euclid did because, for example, he also formalised the property of betweeness, which is, as Pasch pointed out, not proveable in Euclidean geometry.

Tarski was able to prove that Euclidean geometry, formalised as he had done, was:

• complete : every sentence can be proved to be true or false.

• consistent : it was not possible to both prove some sentence and its negation.

• decidable : there is an algorithm that can decide any sentence true or false.

This underlines Megill's point.

However, the fetishisation of axiomatic thinking as being thought that characterises mathematics is simply wrong. And hence also the thought that Godel's theorem somehow prevents the achievement of "absolute rigor". That might have been the agenda of certain mathematicians at certain times, and might have been talked and hyped up by some philosophers who craved "absolute rigor" and thought that mathematics may achieve this. One way that this is stated is that once a theorem is proven, it is true for all time and maybe even outside of time. That sounds rather remarkable and grand with the invocation of eternity and truth. But this doesn't stop whole theories going out of fashion or becoming irrelevant. For example, algebraic geometry based on valuations after the advent of Grothendiecks schemes.

A case in point, physics is inconsistent, even today after four hundred years of development from Galileo. This is acknowledged by everyone, the major inconsistency being between GR & QM. Yet despite its inconsistency, it has achieved great things. More broadly, rational thinking in every day life is often inconsistent and yet the life of the species, species-life, has gone on from strength to strength. Thus inconsistency is no barrier to achievement.

Mathematics is, like all fields and practises, a living and organic expression of the thought and acts of men and women. Absolute rigor, as I have pointed out briefly above, was never its point, but to achieve clarity of thought and expression. A trait that it shares with other fields of human endeavour.

Moreover, there is such a thing as paraconsistent logic where the charge of inconsistency does not explode into triviality - the inconsistency is contained. Whilst what is called classical logic is not paraconsistent, it seems the actual classical thinkers were ahead of us here as Aristotelian logic is paraconsistent. So what Goedel's theorem might be telling us is that we are simply using the wrong underlying logic ...!