What I meant was that I don't believe that you can take the general principle of mathematics and through logical analysis arrive at a foundation of sets, because I cannot see how the concept of 'sets' is contained within the general principles of mathematics. (Whereas I do believe that you can take the general principles of atoms and arrive at a theory of protons and electrons.)
It's a little confusing for you to reply to my words by putting your reply inside my quoted text
in a different color. Extra work for me when I reply. What's wrong with old-fashioned quoting?
Regarding your point in blue, of course I am in full agreement. I have never in my life heard anyone argue that set theory is "is contained within the general principles of mathematics." We have strong evidence to the contrary:
* People have done math for thousands of years. Archimedes, Eudoxus, Gauss, Euler, and Newton were mathematical geniuses who never heard of set theory. Set theory is only a century old; and
* Alternative foundations, several of them, are already beginning their ascendency. In large parts of higher math, specifically algebra and geometry, Category theory
has pretty much replaced set theory (although foundationally-minded category theorists do pay attention to which categories are sets and which aren't, and they have invented some clever patches to make the connection between category and set theory legitimate).
It seems clear to me that you are making a strawman argument, claiming that someone (who?) thinks that set theory is logically entailed by the rest of math, when in fact it is manifestly not.
What continuum? All the evidence points to us living in a quantised world. And in any case I see no problem with a discrete structure modelling analogue data. For example a music cd does an excellent job.
First, there may be evidence but it's in no way conclusive. All we know is that we can't measure below the Planck length. We don't know if that's a problem with measurement or a characteristic of the universe.
But whether the world is discrete or not, humans have an intuition of continuity and a thing called the continuum. A philosopher might well argue that the mathematical continuum is not necessarily the right model of the intuition of the continuum. An intuitionist
would certainly make that argument and many of them have.
But I'm puzzled by something here. I tossed out that example (of the continuum) to GIVE YOU AMMUNITION for your own criticism of set theory! And here you are disagreeing with the very evidence of YOUR OWN THESIS that I'm trying to hand to you.
It makes me wonder why you did that.
it starts with Hume's division of knowledge into the real and the abstract. Mathematics is on the abstract side.
I'm not familiar with Hume, but the rest of what you wrote is quite mysterious. It is the direct opposite of how math works. I can't tell if you are saying this is Hume's account of math, or your account of how you think math IS done or SHOULD BE done. Regardless, what you wrote is so wrong I have to object in detail.
The model for mathematics is then a collection of axioms and methods of inferences.
But Hume lived (according to Wikipedia) from 1711-1776. He knew nothing of machines, set theory, or axiomoatic systems beyond Euclid. I'm a little confused as to whose views you are expressing here.
Then from these axioms an abstract machine is constructed which embodies the axioms and which can then generate 'theorems'.
I don't understand what it means for a machine that "embodies the axioms." Surely I could program the axioms of set theory into a computer, is that what you mean?
The axioms chosen for the system can be somewhat arbitrary.
The axioms of set theory are anything BUT arbitrary. The modern axioms are the result of a painful and difficult process that took place from say 1874, the publication of Cantor's first paper on set theory, to the 1920's or so as Zermelo's axioms came to be widely accepted. Each and every axiom was argued over and dissected by mathematicians and mathematical philosophers. Axioms need to be natural
and embody some intuitive common sense notions such as unions and intersections.
It's true that we could make up any old axioms and play games with them. Chess is one such example. If the knight moved differently or the board was 9x9 instead of 8x8 we'd have a different game, and nobody would care.
But the axioms of math are not arbitrary, they are intended to encapsulate things that we think are true about the world of collections that we call sets.
You used the word "arbitrary" in your previous post and I strongly objected, and now you have repeated that claim without acknowledging that I say you are WRONG. The axioms of contemporary set theory are not arbitrary. They are the result of 40 years of intellectual struggle to pick the right axioms, and in fact set theorists are still looking for the right set of axioms. This fact alone contradicts your claima (repeated twice but never justified) that the axioms are arbitrary.
I apologize if I sound ranty but the axioms are simply not arbitrary. We can talk about this more if you like.
The main requirement of the axioms is that they are consistent
Since it's unknown whether the current axioms of set theory are consistent, I wonder where you got this idea. Gödel showed that the axioms of set theory can not prove their own consistency. The only way to show that they are consistent is to ASSUME stronger axioms, whose consistency is then a matter of question.
with each other to the degree that a machine can be constructed which embodies all the axioms and only those axioms.
Gödel demolished that hope. Although I don't know what you mean by "embodies all the axioms and only those axioms." That phrase doesn't make a lot of sense to me. But if you are saying that an axiom system should be "complete," which means that it any properly formed statement either can or can't be proven from the axioms, then Gödel showed that the ONLY complete axiomatic systems (of suffient power to do number theory) are the inconsistent ones!
Also the machine must be capable of generating theorems. (otherwise it is effectively a null-system).
The third requirement, albeit not a necessity, is that the theorems are 'interesting' to some degree. And certainly for the system of mathematics, the theorems are highly interesting.
Any interesting axiomatic system for math is either incomplete or inconsistent. You are aware of that, right? That's why machines don't do math. We do have early computer theorem proving systems and this is an interesting area of contemporary research. But the human mathematicans are still employed.
It should be emphasised here that the theorems generated are nothing more than strings of abstract symbols that are without 'meaning'. (eg the theorem "2+2=4" is nothing more than an arrangement of the symbols "2", "+" and "4".)
This is the formalist position. But then you haven't explained why the theorems should be "interesting." Why are the rules of math considered ontologically different than the rules of chess? If math were purely a formal game, people still might enjoy playing, but nobody would be arguing that math is central to the nature of the universe.
It do find is awfully strange that you, arguing for intuition, are actually arguing an extreme formalist position that even I don't agree with. Math is NOT just a formal game, that's why its philosophy is so interesting. Nobody claims the universe is made of chess, but Tegmark claims the universe is made of math. Why is that?
The machine can be set to run on its own and generate all the possible theorems of mathematics
Gödel thoroughly destroyed that claim in 1931. Nobody has found a flaw in his argument. On the contrary, we have found NEW proofs of the same fact via computer science (the halting problem) and algorithmic information theory (Chaitin's proof of Gödel's incompleteness theorem).
, albeit not in a finite time.
Now that is really and truly wrong. I hope by the way that you know I'm enjoying our conversation and not piling on in a personal manner. But what you wrote in this post is so contrary to the actual facts about math that I feel compelled to call out each and every misunderstanding, point by point.
It is part of the definition of a formal proof that a proof is a FINITE sequence of deductions. A formal proof could in theory take the age of the universe to write down, but that would still be a FINITE amount of time. I can not imagine what you think a proof is that would require infinitely much time to write down.
What happens in practice is that mathematicians and scientists operate the machine to focus on particular aspects of the system that interest them.
Absolutely and truly false. Math works in the opposite way. Mathematicians are trying to understand the nature of groups, or topological spaces, or differentiable manifolds, and other abstract mathematical objects (which always turn out to be useful to the physicists, contrary to the formalist position).
The pattern is that mathematicians often know what they want to prove, and THEN they write down the axioms that will get them there.
As a striking example, Wiles's proof of Fermat's last theorem is written in the language of modern algebraic geometry, whose axioms go BEYOND standard set theory. It is generally believed that "in principle" Wiles's proof does not actually require the extra axiom (known as the existence of a "Grothendieck universe"). However nobody has bothered to actually write down such a universe-free proof of FLT. Nobody cares about axioms
when they're actually doing math. This is a point lost on many philosophers of math who are stuck in 1900 and who have not yet internalized the post-1950 revolution in the way algebra and geometry are done. Set theory's already dead as a practical matter. I can say a lot more about this if you like.
A simpler example is that Galois never heard of set theory, but he invented group theory in order to show why a 5th degree or higher polynomial doesn't have a formula like the quadratic formula. Mathematicians are not interested in axioms, they're interested in theorems.
In other words: The axioms follow from the theorems
and not the other way 'round like they tell you in naive accounts of the philosophy of mathematics.
The operators of the machine can then take the theorems and 'map' them onto their concepts of the real world.
Really? Why do they do that if the theorems are meaningless? I've never seen anyone try to map the rules of chess to the real world, or the rules of Parcheesi, or the rules of Whist. But we apply math to the real world every day; and most math comes right out of physics at some level. Even set theory
, as the example of Fourier and Cantor shows.
Your naive account of how math works is a strawman. Math doesn't actually work that way.
So for example the symbol '1' would get mapped onto concepts associated with 'one' or 'oneness'; "=" would get mapped onto concepts of "equality" or "equivalence".
How do we map the rules of chess? You are making a formalist argument and then completely contradicting your own position. I hope you don't regard it as unfair of me to point that out. If the theorems are meaningless, why do we think they map onto the real world? If they're meaningful, what makes them so? Perhaps it's the NATURALNESS of the axioms and not their supposed arbitrariness.
It is this useful mapping between the abstract and the real that makes the theorems of mathematics so interesting.
Like I say, you are making both a formalist and a Platonist argument at the same time. I apologize if my post sounds like I'm beating up on your words but you are not stating any kind of coherent position here.
So in this model there is no need for any foundations of mathematics that go any deeper than its axioms.
So what's your object to the axioms then????????? You just argued yourself out of your own position!! You started out by saying you object to basing math on the axioms of set theory; and you ended up by saying that there's no need for any foundation besides the axioms!
tl;dr: Your entire post is a strawman argument describing a theory of math that has absolutely nothing to do with math. And in the end, you argued yourself out of your stated position.