# People who do foundations of maths?

## Do you feel the conjecture is right?

• ### No

• Total voters
18
tgt
Conjecture: Most of the people doing research in the foundations of maths are 'not good enough' for maths.

conjecture here is obviously a joke (but not completely).
not good enough as in feel that maths is too difficult to understand (i.e not clear enough, too abstract).
There's obviously also a personal taste as to why some do foundations and some don't.

I got this idea from Hilbert who thought that abstract mathematics was an elegant way of stating mathematical proofs but that all mathematical proofs could be reduced to a concrete and constructive manner. Godel showed he was wrong but the idea can be applied to wide areas of maths. So in that sense my conjecture seems very true.

maze
Foundations of math seems like a very very difficult and subtle subject from my limited experience with it. I've never heard of anyone saying it was easy.

tgt
Foundations of math seems like a very very difficult and subtle subject from my limited experience with it. I've never heard of anyone saying it was easy.

Don't you hear many people when they don't understand a proof complain it's not clear enough. Then the other person explains it in more detail, in other words (inpolitely) dumbing it down until the person understands it.

Foundations of maths is more then just dumbing down maths but there is an aspect of it to it.

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What exactly do you mean by 'foundations' in this context?

maze
Don't you hear many people when they don't understand a proof complain it's not clear enough. Then the other person explains it in more detail, in other words (inpolitely) dumbing it down until the person understands it.

Foundations of maths is more then just dumbing down maths but there is an aspect of it to it.

I want to make sure we're on the same page here - when I hear "foundations of maths", I think of logic, set theory, category theory, and things like that, and people like Cantor and Godel. Is this what you have in mind?

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Foundations is a hard field -- harder than most, perhaps. I dabble in it, but I don't think I could ever do more.

tgt
What exactly do you mean by 'foundations' in this context?

Foundations of mathematics is a term sometimes used for certain fields of mathematics, such as mathematical logic, axiomatic set theory, proof theory, model theory, and recursion theory.

tgt
Foundations is a hard field -- harder than most, perhaps. I dabble in it, but I don't think I could ever do more.

How come?

tgt
i want to make sure we're on the same page here - when i hear "foundations of maths", i think of logic, set theory, category theory, and things like that, and people like cantor and godel. Is this what you have in mind?

sure.

Homework Helper
How come?

Have you tried category theory or model theory? It's serious stuff. Also all the the large cardinal stuff falls cleanly into foundations, and that's even more heady: I'm just waiting to see how much crashes down the day someone shows a really strong one turns out to be inconsistent.

tgt
Have you tried category theory or model theory? It's serious stuff. Also all the the large cardinal stuff falls cleanly into foundations, and that's even more heady: I'm just waiting to see how much crashes down the day someone shows a really strong one turns out to be inconsistent.

How much of it have you studied? At what level?

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Gold Member
Foundations of mathematics is a term sometimes used for certain fields of mathematics, such as mathematical logic, axiomatic set theory, proof theory, model theory, and recursion theory.

That's the first sentence from the wikipedia page. Do you know anything at all about these fields? If you do, you will know that none of these are 'easy'.

tgt
That's the first sentence from the wikipedia page. Do you know anything at all about these fields? If you do, you will know that none of these are 'easy'.

I'm a beginner but if one was to generalize what Hilbert is describing, it doesn't seem so.

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I'm a beginner but if one was to generalize what Hilbert is describing, it doesn't seem so.

huh?

tgt
huh?

To Hilbert, foundations of maths is making the abstract concrete, which was his programme as well. If we take the foundations of maths as achieving that goal then it would be simpler. Didn't they say that all mathematical proofs can be expanded out to very long if necessary? Is that just one aspect of mathematical logic?

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Gold Member
To Hilbert, foundations of maths is making the abstract concrete.

Where did he say this? Do you have the exact quote?

Staff Emeritus
Gold Member
Have you tried category theory or model theory? It's serious stuff. Also all the the large cardinal stuff falls cleanly into foundations, and that's even more heady: I'm just waiting to see how much crashes down the day someone shows a really strong one turns out to be inconsistent.
I've heard a joke that 0=1 is the last of the large cardinal axioms. (ah, wikipedia is where I saw it)

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I'm a beginner but if one was to generalize what Hilbert is describing, it doesn't seem so.
So, basically, you don't really know what "foundations of mathematics" is, and you don't really know what Hilbert was saying, but you still have the audacity to say "Most of the people doing research in the foundations of maths are 'not good enough' for maths."?

Or are you just hoping that is true so you can do research in the foundations of maths?

Russell Berty
I've heard a joke that 0=1 is the last of the large cardinal axioms. (ah, wikipedia is where I saw it)

:rofl:

Perhaps a weaker axiom would suffice: $$0\approx1$$ or $$0\in0$$

Dragonfall
I focused on set theory during my undergrad years; it's a very difficult subject that is not by any means populated by people who are "not good enough" for other fields. You are very misinformed about foundations, and Hilbert.

tgt
Where did he say this? Do you have the exact quote?

In Shoenfield's Mathematical Logic p3

"Proofs which deal with concrete objects in a contructive manner are said to be finitary... Once the fundamental difference between concrete and abstract objects is appreciated, a variety of questions are suggested which can only be answered by a study of finitary proofs. For example, Hilbert, who first instituted his study felt that only finitary mathematics was immediately justified by our intuition. Abstract maths is introduced in order to obtain finitary results in an easier or more elegant manner. He therefore suggested as a program to show that all (or a considerable part) of the abstract mathematics commonly accepted can be viewed in this way."

What do people make of that? Doesn't the bold suggest the conjecture? Although not all mathematical logic is like that. I've unfairly generalised a bit much.

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In Shoenfield's Mathematical Logic p3

"Proofs which deal with concrete objects in a contructive manner are said to be finitary... Once the fundamental difference between concrete and abstract objects is appreciated, a variety of questions are suggested which can only be answered by a study of finitary proofs. For example, Hilbert, who first instituted his study felt that only finitary mathematics was immediately justified by our intuition. Abstract maths is introduced in order to obtain finitary results in an easier or more elegant manner. He therefore suggested as a program to show that all (or a considerable part) of the abstract mathematics commonly accepted can be viewed in this way."

What do people make of that? Doesn't the bold suggest the conjecture? Although not all mathematical logic is like that. I've unfairly generalised a bit much.

I don't think the bolded section supports your 'conjecture' at all. "Abstract maths" is not "foundations of mathematics", and foundations of mathematics are often/usually infinitary. (Category theory, model theory, the study of large cardinals, etc.) Further, just because a field is introduced to make something easier or more elegant doesn't mean the field is easy or for incompetents.

I also don't think the quote fairly represents Hilbert's position. He was a major proponent of Cantor's program!

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tgt
I don't think the bolded section supports your 'conjecture' at all. "Abstract maths" is not "foundations of mathematics", and foundations of mathematics are often/usually infinitary. (Category theory, model theory, the study of large cardinals, etc.) Further, just because a field is introduced to make something easier or more elegant doesn't mean the field is easy or for incompetents.

I also don't think the quote fairly represents Hilbert's position. He was a major proponent of Cantor's program!

It was assuming that foundationns of maths is conrete compared to maths which is often abstract.

Looking at it another way, from personal experience, I often do not understand abstract maths because I think the terminology and notations are too vague. Foundations of maths introduces less vague notations so things should be easier to understand. Purely in that manner, foundations of maths should be easier and anyone who is willing enough should be able to do it.

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It was assuming that foundationns of maths is conrete compared to maths which is often abstract.
Then you have no idea what "foundations of mathematics" is! Foundations of Mathematics is intimately related to the philosophy of mathematics and is very abstract.

Looking at it another way, from personal experience, I often do not understand abstract maths because I think the terminology and notations are too vague.
Again completely wrong. All terminology and notations in abstract mathematics are excruciatingly precise! They have to be because the are no concrete examples to "point to" as can be done in applications of mathematics.

Foundations of maths introduces less vague notations so things should be easier to understand. Purely in that manner, foundations of maths should be easier and anyone who is willing enough should be able to do it.
This entire thread seems to be based on the fact that you have no idea what "Foundations of Mathematics" means.

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tgt
Then you have no idea what "foundations of mathematics" is! Foundations of Mathematics is intimately related to the philosophy of mathematics and is very abstract.

I was giving an interpretation of what the author was saying in his text.

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It was that fact that you had interpreted it so incorrectly, apparently equating "finitary" with "foundations of mathematics" that made me conclude that you did not know what "Foundations of Mathematics" is. The quote you give says nothing about "Foundations of Mathematics"

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It was assuming that foundationns of maths is conrete compared to maths which is often abstract.

Numerical analysis is concrete; number theory is fairly concrete; foundations of mathematics is very abstract. The term "general abstract nonsense" is used of category theory (a branch of foundations) to poke fun at its abstractness!

You are right, though, that FOM is very precise. It has to be -- unlike in number theory, there's no much intuition to be had at this level!

tgt
It was that fact that you had interpreted it so incorrectly, apparently equating "finitary" with "foundations of mathematics" that made me conclude that you did not know what "Foundations of Mathematics" is. The quote you give says nothing about "Foundations of Mathematics"

I did give an indication that my generalization was wrong in an earlier post.

tgt
You are right, though, that FOM is very precise.

Now my opinion is that FOM should be easier because of nothing else except the higher precision. However, I haven't tried FOM and it might turn out to be just as hard or harder even with this precision (by being more abstract).