[Note: I wrote this long and ungainly post but Arising_uk said the exact same things only in way fewer words].
A_Seagull wrote:
OK, perhaps there do exist formal and explicit axioms that can generate all the theorems of mathematics, but they are not ones that I use or want to use.
Perfectly ok by me. If you've gone from "No foundation exists," to "The standard foundation is not to my liking," you will get no argument from me and perhaps even some points of agreement.
I'm viewing math as a historically contingent human activity. Math was done for thousands of years without set theory. Archimedes and Euclid and Newton and Gauss and Euler never heard of the empty set. Yet they did math.
When I say that set theory is the foundation of math, that's a true statement about the world. I'm not saying that set theory is
necessarily the foundation of math, or the only foundation or the best foundation or even a particularly good foundation; although it's had a successful century-long run. All I'm saying is that set theory
is the foundation of math. We can like it or not but is a fact about the world.
As to how this happened, math had a so-called
fondational crisis around the 1900's, the outcome of which was set theory. I wish I could say a lot more about this but in the interest of brevity I'll leave it to the Wiki page.
Set theory is arguably already losing its foundational importance. One alternative currently being developed is
homotopy type theory, which is influenced by intuitionist philosophy as well as by computer science. (The linked Wiki page isn't very helpful unfortunately). In fact one aspect of HoTT as they call it is computer verification of proofs. Clearly this will be a big thing in the future. You remember that detailed proof you challenged me to do. There's an app for that!
Another modern development is
Category theory, which is based on philosophical structuralism. Objects are not characterized by what they are; but rather on their relationships with other objects. Most modern algebra and geometry is done in the framework of Category theory these days.
If you don't like the current state of things, that's perfectly fine. However if you want to know what the current state of foundations is, it's set theory. For no other reason than that's the way it turned out to be. Today. Next year, next decade, next century? Human thought moves ever forward.
A_Seagull wrote:
Why invoke set theory as foundational? - it just seems to introduce unnecessary complication.
Lot of very good reasons. Goes back to the rigorization of calculus, the renewed interest in foundations after the discovery of non-Euclidean geometry, the discovery of set theory by Cantor, the discovery by Russell that Cantor's set theory was inconsistent, the growing need to get everything cleaned up. A lot of fascinating history and philosophy in there from say 1870 to 1940, say
A_Seagull wrote:
Why Peano's axioms? - If you didn't know that they were supposed to generate the natural numbers you wouldn't realise it from the axioms themselves.
Some guy named Giuseppe Peano wrote down some axioms in 1989. The idea was in the air. This Wiki article gives some of the history.
https://en.wikipedia.org/wiki/Peano_axioms. Again it's a matter of historical contingency. Peano was born, became a math professor, people were trying to axiomatize the natural numbers, he worked on the problem. On some other planet they still have the natural numbers (presumably) but not the same set of axioms, if they even have axioms. Platonic math and human-made math, two different things. Except that maybe Platonic math doesn't exist and historically contingent math is all there is. Philosophers like to kick that one around.
A_Seagull wrote:
What was the point of Russell and Whitehead taking several volumes of writing to 'prove' that "1+1=2"? To prove anything in a deductive system you need a foundation, an original axiom if you like, so why didn't they just start with "1+1=2" as their original assumption? Then the job would have been done in one line!
It was in the air. There were all these mathematical and philosophical problems around. Russell went after the problem. It's a singular work, very difficult. I have no idea why Russell did what he did.
But I will disagree with you about making 1 + 1 = 2 an assumption. You haven't told me what the symbols mean. And around 1900 it became really important to get things nailed down to make sure mathematical reasoninig was valid. Nobody knew what the right principles were. It was all part of a long historical process.
Remember that during this time, relativity and the quantum theory were being discovered. You could ask why those theories popped up and why they took the form that they did. All good questions in the history of thought.
A_Seagull wrote:
It seems to me that all these attempts to create a formal axiomatic system for mathemnatics are trying to prove something from nothing. But of course you can't do that. So instead, the assumptions are hidden within their axiomatic assertions.
The assumptions aren't hidden. For the first time they are clearly revealed. Now we know how to make sense of the infinitary operations of calculus. We know what are the underlying principles of number theory. We understand the power and limitations of mathematical logic. All this had to be worked out explicitly by smart but fallible people over many decades.
A_Seagull wrote:
Even if your axioms do provide a complete axiomatic system for mathematics, can you be sure that there is not a simpler set of axioms that do the same job but more efficiently? I doubt it.
Ah the axioms are NOT complete! That's what Gödel showed in the 1930's: that no matter what axiom system you pick, there will always be mathematical facts that are true but that can't be proved in that system of axioms. This was a big deal, one of the profound intellectual shocks of the twentieth century. And Gödel's work came directly out of Russell's. In fact Gödel was working in Russell's system to show that axiomatic systems were inherently incomplete.
A_Seagull wrote:
So that is why I prefer to consider the axiomatic foundations for mathematics to be implicit rather than explicit. All an axiomatic system for mathematics needs to do is to define the relationship between the symbols of mathematics and how the true theorems of mathematics can be derived from these relationships.
If you like the relationship-oriented approach, structuralism and Category theory are for you.
https://en.wikipedia.org/wiki/Structura ... thematics) [link doesn't catch closing paren, need to copy/paste]
I hope I've at least explained what I mean when I say that set theory is the foundation of math; even if it's not philosophically satisfying to you. It's just the way it is. Everyone who studies math works in the framework of set theory. In the future, who knows. It'll all be done by computer, even the proofs. The state of math is always changing because math is a historically contingent activity of humans.