Wikipedia:Reference desk/Archives/Mathematics/2022 May 5

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May 5

Axiomatics

Is there such a thing as axiomatics as an independent academic discipline? Are there real mathematicians or researchers working in this field?--2A02:908:422:9760:0:0:0:D631 (talk) 18:01, 5 May 2022 (UTC)

Inasmuch as scholars have worked on developing axiomatic systems, it has (to the best of my knowledge) always been in the context of some specific field, like geometry, algebra, set theory, probability theory, ... – and even of mathematical models of physical theories, such as thermodynamics and quantum field theory. I don't think this kind of work is a discipline of its own, abstracting from the fields such work is applied to. I suppose that the ability to axiomatize a non-trivial theory requires a thorough insight that only develops by doing significant work within that theory. I expect it helps to have a strong background in model theory.  --Lambiam 19:41, 5 May 2022 (UTC)

Axiomatic system has some material on the study of axiomatic systems in math and logic. I am sure there are some researchers in metamathematics who consider the study of axiomatic systems in mathematics as part of their discipline. --{{u|Mark viking}} {Talk} 20:17, 5 May 2022 (UTC)

I would tend to think this falls in the category of either logic or metamathematics. D A Hosek (talk) 04:45, 6 May 2022 (UTC)

• Check out formal science. There are conferences devoted to the formal sciences, such as FotFS; attendees tend to come from many different disciplines. It can be seen as applied logic. It can be more or less mathematical. — Charles Stewart (talk) 05:22, 6 May 2022 (UTC)
• Perhaps something like Russell & Whitehead's Principia Mathematica or the work of Kurt Gödel, David Hilbert, etc. The early 20th century was a HUGE time for foundational mathematical thinking, and a lot of the work of those and other affiliated mathematicians at the time could be said to be in the field of axiomatics. They were trying to actually find the fundamental axioms of mathematics; i.e. how basic could you get. Principia Mathematica famously took 340 pages of dense symbolic logic to prove 1+1 = 2, with the goal of doing so using as few axiomatic assumptions as possible. --Jayron32 11:56, 6 May 2022 (UTC)

I think you may be looking for philosophical logic. This is a somewhat wider field than mathematical logic, which crosses between math and philosophy, and can be studied in both math and philosophy departments. 2601:648:8202:350:0:0:0:4671 (talk) 20:31, 9 May 2022 (UTC)

I think this is not right. Plenty of axiomatics happens outside philosophical logic, and a fair bit of philosophical logic is not axiomatic in nature. — Charles Stewart (talk) 15:06, 10 May 2022 (UTC)

The important thing about axioms is that they are irreducible. Axioms are the "fundamental particles" of logic, upon which logical systems, (of which mathematics is but one) can be built. Axiomatics as a field of study is thus the equivalent of "particle physics" or "quantum mechanics" for something like mathematics. As I mentioned above, plenty of thinkers from 100 years ago were working in the field, attempting to determine the fundamental nature of mathematics. I personally find the conclusions of Kurt Gödel to be the most interesting; his incompleteness theorems are essentially the mathematical equivalent of quantum mechanics: they establish the limits of mathematical systems in the same way that QM has, at its core, a certain level of unknowability. If someone wants to find modern mathematicians working in the same field, I would start with people like Gödel and Hilbert and Russell and perhaps Emmy Noether and definitely Zermelo and Fraenkel and then find people who continued their work. --Jayron32 15:41, 10 May 2022 (UTC)

It's kind of hard to be sure what exactly you're asking. Mathematical logic is a very broad field with lots of subdisciplines. Two of them are proof theory, which is most concerned with how axioms are put together to derive other statements, and model theory, which concerns the properties of structures satisfying certain axioms. Mathematical logic is traditionally also taken to include computability theory and set theory, the last being my academic field.

However neither of those is usually referred to by the specific term "axiomatics". If you are asking whether there's a field that is usually called by the specific word "axiomatics", I would probably say no, though I wouldn't be shocked to hear that some math department somewhere used it.

One meaning the term could have in some possible world would be investigations into where axioms come from. Are they self-evident? Chosen arbitrarily? Obtained empirically or inductively? Chosen according to the beauty of their consequences? (My personal answer would be a combination of 1 and 3, counting 1-based, but that's not important right now.)

These are fascinating questions, and I don't really know of a standard name for this line of inquiry, which is sort of surprising, or maybe it just means I'm slow today. If you're interested in this aspect, you could do worse than to check out two classic papers by Penelope Maddy, called Believing the Axioms, I and Believing the Axioms, II. They were both published in the Journal of Symbolic Logic sometime in the 1980s, I think (by the way, Maddy has changed her view since then, but the papers themselves are still very much worth reading). They should be available on JSTOR. --Trovatore (talk) 19:11, 10 May 2022 (UTC)