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Reply 2060

III Algebraic Geometry is advised to be one of the best Part III courses to do in the third year, because it brings a lot of less advanced material together as well as providing a platform for more advanced material.

This year it was more or less the same as Part II Algebraic Geometry. It depends on the lecturer. Commutative Algebra is a surer bet. If you're geometrically inclined, take either Part II or Part III Algebraic Topology – the ideas are really foundational.

The same is also said about Part III Category Theory.

You could go to that in first year and it wouldn't be any harder to understand. The only difference would be a lack of examples and motivation.

Are there any other pure Part III courses - e.g. in analysis? - that are similar in this way?

There's hardly any analysis in Part III...

Reply 2061

around

Original post by Zhen Lin

You could go to that in first year and it wouldn't be any harder to understand. The only difference would be a lack of examples and motivation.

wait...

Reply 2062

glowinglight

Original post by Zhen Lin

Wasn't sheaf cohomology in Part III Algebraic Geometry this year? It's not in Part II. I will have a look at Commutative Algebra.

You could go to that in first year and it wouldn't be any harder to understand. The only difference would be a lack of examples and motivation.

I find more motivation makes ideas easier to understand. Examples, not always so much.

There's hardly any analysis in Part III...

Where did it go? :smile:

Did they move courses to the CCA? What's an analysis head supposed to do?

Reply 2063

Zhen Lin

Original post by around

wait...

The proofs are really easy, with exceptions I can count on one hand. It's very much "there's only one thing you can do, and it works". Some take this as a sign that there isn't any real content in the subject, and this is where the phrase ‘general abstract nonsense’ comes from.

I have mixed feelings about whether it's better to see category theory first or to see the rest of mathematics first. On the one hand, it would be good to know what a universal property looks like so that you can recognise one as soon as you see it. On the other hand, it is probably quite hard to understand what a universal property is unless there are a half-dozen examples to retroactively recognise.

I'd love to carry out an experiment to get some data about this...

Original post by glowinglight

Wasn't sheaf cohomology in Part III Algebraic Geometry this year? It's not in Part II. I will have a look at Commutative Algebra.

Very little. It was only Čech cohomology, and it was taught very very quickly. Derived functor cohomology showed up in Algebraic Topology, but was equally rushed. I repeat, it depends on the lecturer. The only course which treated sheaf cohomology properly was Complex Manifolds.

I find more motivation makes ideas easier to understand. Examples, not always so much.

Examples and motivation are one and the same in category theory. Remember, the subject is about finding what is common between different areas of mathematics.

Where did it go? :smile:

Did they move courses to the CCA? What's an analysis head supposed to do?

Well, if you're willing to accept numerical analysis or theoretical probability as analysis, then there's that...

(edited 12 years ago)

Reply 2064

around

Original post by Zhen Lin

The proofs are really easy, with exceptions I can count on one hand. It's very much "there's only one thing you can do, and it works".

You need a lot of maths behind you before 'doing the obvious thing' proofs stop seeming stupid and start making sense.

(edited 12 years ago)

Reply 2065

Zhen Lin

Original post by around

Original post by Zhen Lin

The proofs are really easy, with exceptions I can count on one hand. It's very much "there's only one thing you can do, and it works".

You need a lot of maths behind you before 'doing the obvious thing' proofs stop seeming stupid and start making sense.

No, not at all. There's no "real" mathematics happening – you have to see it to believe it! (For example, the fact that limits are unique up to unique isomorphism when they exist, or the fact that pasting two pullback squares along a common edge makes the outer rectangle into another pullback diagram...)

Reply 2066

around

Original post by Zhen Lin

It's precisely this reason why first years aren't the most appropriate people to show category theory to.

Reply 2067

Zhen Lin

Original post by around

It's precisely this reason why first years aren't the most appropriate people to show category theory to.

*shrug* The sooner they learn that much of mathematics is just a game with funny rules...

Reply 2068

trollbuster

Original post by Zhen Lin

*shrug* The sooner they learn that much of mathematics is just a game with funny rules...

...said the ghost of Wittgenstein as it took over another young man's soul...
...and what's a 'game'?...
...so how much have you actually said, that isn't tautological?...
...is a mathematical object a game with funny rules?...
...what's the difference between mathematical objects and rules for combining them?...
...bear in mind that a difference doesn't require a division...grey areas are allowed...
...you need more tension and dynamic in your thought...

Staying on the facts, many of the objects used in the category theory examples are not properly gettable until after one has assimilated most of the big ideas one learns in Parts IA and IB.

Reply 2069

Zhen Lin

Original post by trollbuster

1. This is Wittgensteinism? I had no idea.
2. Does it matter?
3. It's all tautological and relative to the choice of foundations.
4. Mathematics is a game; mathematical objects are the things which appear in it.
5. A mathematical object is like a word: it has a purported meaning; continuing the analogy, the rules for comining them is like syntax. The difference between syntax and semantics is well-understood.
6. There are parts of mathematics which are more "real" than others.
7. Perhaps you need to think about mathematics more...

Staying on the facts, many of the objects used in the category theory examples are not properly gettable until after one has assimilated most of the big ideas one learns in Parts IA and IB.

And your reasoning is? Like I said, there is no data on how well this would work. Even my set-theorist friend thinks that there should be an undergraduate category theory course. Lawvere and Schnauel have produced a textbook aimed at teaching high school students category theory.

Here's my view: I don't think of myself as being particularly clever. I credit my ability to do mathematics to having had an early interest in unifying ideas (I heard about category theory when I was in school. :tongue:

): when one knows the common thread underlying different manifestations of the same abstract idea, then there is less to remember and less to understand. It also helps motivate definitions: for example, the only way to explain why the box topology is not the right definition for the product of infinitely many topological spaces is to say that it has the wrong universal property. Similarly, the reason why the infinite product of modules is not the same as the infinite direct sum is that they have different universal properties. etc.

Reply 2070

trollbuster

Original post by Zhen Lin

1) yes - the aphorismised "game" formulation comes from several decades of somnambulistic passing down in an East Anglian town of the 'ideas' of Wittgenstein, I'm afraid

2) if you say mathematics is a game in the belief that it's a profound insight, then surely it does matter what a game is - I was alluding to Wittgenstein's attempt to build on the problems involved in defining the word "game", which attempt to me has long seemed unhelpful

3) I disagree; but if I'm wrong, what weight would your 6 have?

4) it gets nowhere to say mathematics is a game; however, the idea that mathematical objects are the objects that appear in it - you can get somewhere with that, if you put it together with the idea of rules and abstractification and concentrate on the tension and dynamic; my 5,6,7 were the most important in my post

5) if this analogy led anywhere, we might as well talk about language and ignore mathematics

6) maybe so - but why base the direction of your thoughts about profundities on concepts to which you are still giving scare quotes? :smile:

I agree it could be done and that it could be useful. But it could not just be added in to today's Part IA. This is because it would require an understanding of more classes of objects than just the objects taught in IA; and even introducing just a few more which follow easily from those would not be sufficient. IA would have to be completely reorganised. And doing it to teach people that mathematics is a "game", rather than whatever they'd otherwise think it was, could not be a serious organising idea motivating such a reorganisation (and anyway is a wrong idea.)

While I admire your enjoyment of category theory, category theory isn't such a qualitative step forward as you seem to think. You are coming close to depicting it as the last word. It isn't. Nothing is. Everything is movement. Heraclitus had it right. You have been learning about new mathematical objects, new ways for dealing with old objects - often unifying ways which enable a more profound and simple understanding of the old objects - and how to view such new ways as themselves objects, and then going round and back and up and forward, for years. That is what mathematics is about, in my view.

Here's my view: I don't think of myself as being particularly clever. I credit my ability to do mathematics to having had an early interest in unifying ideas (I heard about category theory when I was in school. :tongue:

What do you think "two" is, or "square" is, if they aren't unifying ideas? :smile:

when one knows the common thread underlying different manifestations of the same abstract idea, then there is less to remember and less to understand.

I would agree that a lot of time is wasted in teaching mathematics to children slowly, and that it could be done much better; more often than not the teachers have no clue about what we have just been talking about!

Reply 2071

Zhen Lin

Original post by trollbuster

1) yes - the aphorismised "game" formulation comes from several decades of somnambulistic passing down in an East Anglian town of the 'ideas' of Wittgenstein, I'm afraid

2) if you say mathematics is a game in the belief that it's a profound insight, then surely it does matter what a game is - I was alluding to Wittgenstein's attempt to build on the problems involved in defining the word "game", which attempt to me has long seemed unhelpful

I never claimed it was a profound insight. Moreover, I did not claim that mathematics was meaningless as a result. Rather, mathematics as practised is a game played by professionals: one mathematician is simply trying to convince another that what he says is valid. This part of the game is informal, as is any game of wits. The formal part of the game is otherwise known as symbolic logic.

As for the meaning of mathematics, I hold that it is something amazing that mathematics does seem to approximate the real world. To some extent this is because we have chosen the rules of the game to make it work – but, just as with science, we must be prepared to change the rules when predictions don't match experiment.

3) I disagree; but if I'm wrong, what weight would your 6 have?

The point is that mathematics does not inherently have value. What value it has comes from empirical evidence that it works: so it is a contingent belief. If one day it should be discovered that there are phenomena not adequately explained by mathematics, so much the worse for mathematics.

5) if this analogy led anywhere, we might as well talk about language and ignore mathematics

Human language is a little too imprecise and far too complex to play with. Mathematics is much easier.

6) maybe so - but why base the direction of your thoughts about profundities on concepts to which you are still giving scare quotes? :smile:

Because, in my view, nothing in mathematics is "real". The parts which are more "real" are the parts which are more strongly rooted in empirical observations. I believe that 2 + 2 = 4 because this is something that agrees with what I see in real life, not because I can prove it. But my only reason for believing that non-measurable subsets of the real line "exist" is that I can prove that they exist – and even then only under some assumptions. In other words, what is more "real" for me is what I find intuitive – and, honestly, what is the point of mathematics if it matches neither expectation nor experiment?

Please do explain what classes of objects you would introduce in order to teach category theory. You seem to have spent some time thinking about this, why not share?

While I admire your enjoyment of category theory, category theory isn't such a qualitative step forward as you seem to think. You are coming close to depicting it as the last word. It isn't. Nothing is. Everything is movement. Heraclitus had it right.

Amusingly, Lambek quotes Heraclitus in exactly the same way, to explain the view that it is the functors between categories that are important, not the categories themselves. (Of course, the quote was contrasted against a hypothetical Pythagoras, "everything is categories". Note also that this is one level up from the usual category-theoretic view that it is the morphisms between objects that are important.) He also cites the influence of "the unity of opposites" on Lawvere's view that "adjunctions are everywhere".

I don't claim category theory is the last word. Nor do I claim it is useful in all areas of mathematics. But I find it highly unlikely that the Grothendieck school of algebraic geometry would have gotten as far as they did without category theory – they certainly invented enough of it on their own.

You have been learning about new mathematical objects, new ways for dealing with old objects - often unifying ways which enable a more profound and simple understanding of the old objects - and how to view such new ways as themselves objects, and then going round and back and up and forward, for years. That is what mathematics is about, in my view.

Yes. And category theory has been immensely helpful in that regard. Why merely assert that two concepts are similar when one can exhibit them as two concrete instances of the same abstract notion? Or make vague statements like "the isomorphism between a finite-dimensional vector space V and its double dual V** is natural, but the isomorphism between V and V* is not" when this can be made precise and proven? (But even here, where category theory made its debut, it is not (yet) the last word. There are constructions which are "obviously" canonical, like the group of automorphisms of any object, but which are not even functorial. And it is not clear how to deal with things which are unique, but only up to non-unique isomorphism – like the algebraic closure of a field. More work is needed in this area.)

What do you think "two" is, or "square" is, if they aren't unifying ideas? :smile:

Why stop there, when we can go so much further?

Reply 2072

trollbuster

Hi ZL, you are expanding the field of discussion somewhat, and although I do not wholly share your views about the practice, meaning, and value of mathematics, I haven't really got time to go into those things now - except just to make the point that you do not need empirical evidence that a mathematical proof works, and when you get more and more abstract, such evidence is often not available anyway. In fact that is even so with less abstract ideas such as the infinite size of the set of primes, for which I'm unaware of any empirical evidence. So does that fact have no 'value'? (Engineers need not answer.)

Empiricism is very 'Cambridge', or 'Anglo-American' if you like. Not my cup of tea at all. Maybe even in its own terms, it's had its day where intellectual advances are concerned? Notice that the Ecole Normale Superieure has absolutely kicked Cambridge's butt since 1998 where Fields Medals are concerned. Mathematics is theoretical, not empirical. Interestingly, Wittgenstein, although he based a lot of his stuff on mathematics, did not actually make any advances in mathematics. Perhaps he was just enough of a mathematician to be rated by philosophers, and just enough of a philosopher to be rated by mathematicians? There are others about whom a similar point could be made - it is common for those who have wholly unfairly reached acclaim as 'great minds'.

Original post by Zhen Lin

Why merely assert that two concepts are similar when one can exhibit them as two concrete instances of the same abstract notion?

Who on earth was saying one should?

Yes. And category theory has been immensely helpful in that regard.

But you should realise that things you learnt beforehand have been too, and so will things you learn afterwards be.

This was the intended point of my asking:

What do you think "two" is, or "square" is, if they aren't unifying ideas?

and you must have insufficiently grasped the point if you then tritely ask:

Why stop there, when we can go so much further?

Nobody is saying stop anywhere. Exactly the opposite is the point.

But I feel you were talking up category theory too much (implicitly in relation to what you already knew), and it's doing that that would present a danger of stopping!

Thanks for the reference to Lambek, for which I'm grateful.

You may also find this document useful, penned by an old Trinitarian :smile:

Reply 2073

Zhen Lin

Original post by trollbuster

You seem to have misunderstood me. Mathematical proofs "work" because they are valid plays in a game. There is no need to empirically demonstrate this, and yes, indeed, my point is that empirical evidence is simply not available in abstract evidence. What needs empirical demonstration is that mathematics has anything to do with real life.

As for the infinitude of primes: what the proof really shows is that we may construct arbitrarily large prime numbers, and even better, the proof is entirely finitistic and effective. (How could it not be? It was known even in Euclid's time, well before we developed the sophistication of modern mathematics.) It is very much empirically refutable – when interpreted in that light. So it already has some value, even without knowing that large prime numbers have practical applications in cryptography. On the other hand, if you interpret it as the statement that "there is a bijection between the set of natural numbers and the set of prime numbers" or something equally abstract, then I'm afraid I have to agree that this is not an empirically refutable claim, because there are no infinities in reality.

Empiricism is very 'Cambridge', or 'Anglo-American' if you like. Not my cup of tea at all. Maybe even in its own terms, it's had its day where intellectual advances are concerned? Notice that the Ecole Normale Superieure has absolutely kicked Cambridge's butt since 1998 where Fields Medals are concerned. Mathematics is theoretical, not empirical.

Verily. I gladly do theoretical mathematics. But I don't pretend it's of any real value.

But I feel you were talking up category theory too much (implicitly in relation to what you already knew), and it's doing that that would present a danger of stopping!

And I feel that you don't understand the history or the content of the subject very well, if you think that taking it seriously will any way hinder the "unification" of the parts of mathematics it touches. It's not a finished programme: it's not even a hundred years old yet. Sadly, unification is not a programme that many people care about, and many people use "abstract nonsense" literally as a disparaging remark about category theory. You'll understand if I'm defensive about the subject.

There are certainly parts of pure mathematics that will probably be forever out of the reach of category-theoretic methods – combinatorics and hard analysis come to mind – but I'm not interested in those areas (and it doesn't help that I'm not any good at them) and I would be happy to let the professionals keep going as they always have.

You may also find this document useful, penned by an old Trinitarian :smile:

I'm not going to immediately find anything useful if I'm not told what it might be useful for... (and I certainly don't care whether it was written by an old Trinitarian or a beggar child on the street; ideas are to be judged on their own merits, not where they came from.)

(edited 12 years ago)

Reply 2074

trollbuster

Original post by Zhen Lin

As for the infinitude of primes: what the proof really shows is that we may construct arbitrarily large prime numbers, and even better, the proof is entirely finitistic and effective. (How could it not be? It was known even in Euclid's time, well before we developed the sophistication of modern mathematics.) It is very much empirically refutable – when interpreted in that light. So it already has some value, even without knowing that large prime numbers have practical applications in cryptography. On the other hand, if you interpret it as the statement that "there is a bijection between the set of natural numbers and the set of prime numbers" or something equally abstract, then I'm afraid I have to agree that this is not an empirically refutable claim, because there are no infinities in reality.

It would be empirically irrefutable if it weren't true (at least in principle), but we know it's true without reference to any empirical evidence, and we also know it cannot be empirically verified.

Original post by Zhen Lin

It's true that I don't know the history of category theory very well, but nowhere did I criticise you for taking the subject seriously. I was suggesting you were overdetermining its importance in relation to other abstract unifyings both lower down and higher up - and what's really important is the movement. An appreciation of this helps with learning any subject.

Original post by Zhen Lin

(...)and many people use "abstract nonsense" literally as a disparaging remark about category theory. You'll understand if I'm defensive about the subject.

OK! None of that attitude from me, though :smile:

Original post by Zhen Lin

I agree. I put the smiley to emphasise that the 'Trinitarian' comment was intended to be humorous! The author left Trinity and Cambridge before taking his finals, and later the porters were told to look out for him and not to let him in. He said he wanted his sarcophagus kept over the gate. This may or may not shed light on contradictions in his attitude...or in the rest of the world, bar his attitude :smile:

I think the referenced text is useful because it helps with grasping the fact that the hierarchy of abstract unifications has no limit, and also helps deepen understanding of the 'phases' of the learning process.

Edit:

ideas are to be judged on their own merits, not where they came from.

And dismissed on their demerits :smile:

(edited 12 years ago)

Reply 2075

Mr Dactyl

Original post by trollbuster

East Anglian town

It's a city.

Besides, "philosophy of maths is about as much use to mathematicians as ornithology is to birds", although I never remember whether it was Dirac or Feynman who said that.

Reply 2076

Zhen Lin

Original post by trollbuster

It would be empirically irrefutable if it weren't true (at least in principle), but we know it's true without reference to any empirical evidence, and we also know it cannot be empirically verified.

I don't understand what you're saying. Euclid's proof of the infinitude of the primes says something very precise about where I can find prime numbers: everything has explicit bounds. Indeed, the proof tells us that there is at least one prime in the set

\{ n + 1, \ldots, n ! + 1 \}

for every natural number n. It is possible, in principle, to refute Euclid's proof by exhibiting an n for which this fails. But I agree it cannot be empirically verified. Other formulations of the infinitude of the primes are irrefutable: for example, if we take "there are arbitrarily large primes" literally, then neither empirical refutation nor empirical verification are possible.

Now, as for things being "true", I think we should avoid that minefield altogether...

I can't say I felt much of a desire to read the whole thing: I simply did not see what point he was trying to make. Would that philosophers be so kind as to state their thesis at the beginning of their arguments...

Reply 2077

lopterton

Original post by lopterton

Does your DOS have much role when you do Part III? Does your college?

Can someone help with this? I mean what role does your DOS have in practice in Part III? My DOS is a complete idiot, jerk, and wrecker, but is considered to be uncriticisable by the tutors at my college, including my own tutor, who have pooh-poohed complaints from a few of us and have refused to sack him or move us to another college. If he'd have a big role during Part III, I won't do Part III - I'll do a Master's somewhere else instead.

Reply 2078

nuodai

Original post by lopterton

Their role is less important, but it's still 'officially' the same -- they're the person who sees your CamCORS reports, who you meet at the start and end of each term, and so on. But you'll also have a departmental contact who is likely to be more useful for things relating to Part III specifically; your DoS will probably have quite a diminished role in this respect. See Section 8 of the Part III Handbook.

Reply 2079

lopterton

Original post by nuodai

Hi nuodai. Thanks for this. I've read the Handbook and everything I can find. The issue is still very much "on the edge" for me. I think I'd do fine with the departmental contact, and welcome a diminishing in my DOS's role - but how much scope would my DOS still have for messing things up if he wanted to? There probably isn't a good answer to that! If he did his usual stuff, though, since a DOS is a college matter, I'd be complaining to the people I've already complained to (i.e. at my college), when I got pooh-poohed. Rather than face this, I'd prefer to do a Master's somewhere else. On the other hand, if I do Part III, maybe my DOS would stop caring about exerting an influence, and leave most of the job of any required involvement relating to my maths studies to my departmental contact. (In which case he's very welcome!) There's hardly any way to tell, so it's a difficult one!

Maybe one thing to ask is - how much work does a DOS have to do (other than seeing you at start and end of term) when they've got a student in Part III? If the answer is 'not much', this may bode well :-)