Yes, that’s true. Sometimes I think of algebraic geometry as the quest to develop ‘double vision’, where everything in algebra (well, at least the algebra of commutative rings) can be seen as geometry, and everything in geometry (well, at least some kinds of geometry) can be seen as algebra. And despite the parenthetical comments, mathematicians are always working to expand the scope of this ‘double vision’. They’re trying noncommutative algebraic geometry, absolute geometry, derived algebraic geometry, and much more.

]]>Hi, Giampiero! You wrote:

From an engineer’s perspective, I guess that the less one’s field is “applied” the harder might be to visualize stuff, since one tends to lose the all-important anchoring to physical (or at least physically inspired) models.

I would refine this a bit: you seem to be conflating “applied”, “easily visualized” and “physical”. There are branches of pure math that are very easy to visualize: for example, branches of geometry, especially geometry that involves 2- or 3-dimensional shapes one can draw. There are very applied branches of mathematics that are hard to visualize, at least for me, like cryptography. And “physical” seems to be a third axis.

But I agree that there are lots of people who more easily approach subjects that are applied, visualizable *and* physical. All these are different aspects of being “concrete” rather than “abstract”, I guess.

In math departments there are lots of people who find applied and physical problems *harder*. Some of these people still like visualize problems: for example, geometers and topologists. Others shun visualization: for example, some algebraists.

I tend to think there are 3 methods of understanding math, and I try to keep getting better at all of them: visualizing, calculating, and verbal reasoning.

While i’m here i also can’t resist to comment about Wigner’s “unreasonable effectiveness” of mathematics in describing the universe: maybe i am being too much of a reductionist but shouldn’t it be true that in any universe in which stuff does not happen randomly, observational data can be compressed, and therefore, you can describe them with mathematical laws?

I can imagine, or at least I *think* I can imagine, a universe we could live in that had lots of patterns—but few or none holding with mathematical precision. For example, maybe gravity would attract masses and decrease with distance, but vary in strength slightly in an unpredictable way depending on time and place.

I can also imagine—I think—a universe governed by mathematical laws, but only laws that don’t use very *deep* mathematics. No Riemannian geometry, no gauge theory, no functional analysis. Why, for heaven’s sake, should atomic nuclei be held together by a for described by an SU(3) gauge theory? Wouldn’t some simple sort of sticky behavior be enough?

These are the things that make me feel mathematics is “unreasonably” effective in describing our universe—that is, apparently better than it needs to be.

I hope we will someday know a reason for this phenomenon, so I’m putting “unreasonable” in quotes: that’s just the word Wigner used for it. And I don’t think about this much anymore… but it did help launch me on the path of wanting to learn the fundamental laws of physics, back when I was in college. After pondering this stuff a while, I decided that *first* we should learn what the laws of physics are, and *then* think about “why” they are that way.

Thank you so much for this fascinating post. Reading it (together with all the replies) was a real pleasure! From an engineer’s perspective, I guess that the less one’s field is “applied” the harder might be to visualize stuff, since one tends to lose the all-important anchoring to physical (or at least physically inspired) models. So i can conjecture that going from mechanical engineering to electrical engineering to physics to applied math to pure math it might become harder to visualize things. While i’m here i also can’t resist to comment about Wigner’s “unreasonable effectiveness” of mathematics in describing the universe: maybe i am being too much of a reductionist but shouldn’t it be true that in any universe in which stuff does not happen randomly, observational data can be compressed, and therefore, you can describe them with mathematical laws? I’m sure that this must be a thought that has already occurred to you, so i guess i’m missing something. Or maybe it’s a little bit like a spiritual experience to look at the universe and wonder in awe how can everything be possibly described by a bunch of simple and elegant (of course after you understand them) laws …

]]>i would be presented with something like Schrodinger’s equation and told to solve it. I told them its already solved. It has an = sign. They wanted me to rearrange it.

Pascal was like Gorondiek another french mathematician as was Galois and cedric Villani. (my first required presentation in undergrad was to prove the boltzmann’s h-theorem which Villani got his fields medal for a related topic—i couldn’t get through it so i proved it my own way–my professor fell asleep during that presentation on purpose–he didn’t like me ). .

The teacher who prescribed Pascal to me also tried to flunk me (said i had poor class attendence). I met some grad students who were into differential geometry, related to algebraic—descartes sort of turned geometry into algebra. They were all going to Hollywood to make graphics for movies of the kind i don’t like.

the relationship between algebra and geometry has always interested me though i dont know much about it. my linear algebra teacher gave me this long review article from AMS 1980 by G S Mackey on ‘harmonic analyses as the exploitation of symmetry’ (Hermann Weyl also has an old book on this–the relation between pictures and math). That teacher also tried to flunk me due to poor class attendence.

Except for a rare few, I think good mathematicians learn each concept along with a battery of examples that illustrate that concept—and also counterexamples, things that come close but *fail* to be examples of that concept.

E.g., I don’t think anyone can claim to really understand the definition of a “Hausdorff” topological space if they don’t know a topological space or two that’s not Hausdorff.

Textbooks sometimes leave it to us to find these examples and counterexamples ourselves. They may present definitions and immediately move on to theorems using those definitions. But that means we have to pause and collect some examples and counterexamples ourselves.

Also, whenever you read a theorem, you should think about how dropping any hypothesis would let in examples of things that don’t obey the conclusions of that theorem.

As you collect your gallery of examples and counterexamples, you can test them for each new property you learn. For example, if you’re studying rings you need to keep the ring of integers (and a bunch of others) by your side. Then, if someone defines a “Dedekind domain”, you should instantly check to see if the integers are a Dedekind domain. First, because this is good to know. And second, because figuring it out will give you some insight into what a Dedekind domain is really like!

I guess what all this amounts to is that reading mathematics is a very active process of engaging with the text and testing out what it says — not just reading and remembering.

]]>Wolfgang wrote:

I thought about it a while, and I think I realize that those mathematical problems I had to deal with so far were fun for me, whenever there was a strong visual aspect present, either very direct as it was a problem one could graph, or more indirect in being able to create suitable mental images about the process, however crude they may have been. I guess I am a visual person, which makes purely algebraic texts and the typical definition-theorem style of math papers a challenge for me.

I began my math career as a very visual thinker. I wasn’t bad at manipulating symbols according to rules, but to feel l really understood something I needed to have a mental image of it. I gradually extended my visualization ability to handle higher-dimensional or infinite-dimensional spaces, where I wound up sort of ‘faking it’, visualizing in 3 dimensions but imagining it as higher-dimensional and knowing what corrections are needed. I was also okay with general topological spaces (where in general vague pictures of blobs are what you need) and other abstractions that were still based on some notion of ‘space’. But I was not very good at abstract algebra.

I finally got over that deficiency in a couple of ways. First, I learned how to visualize algebraic structures like groups and rings. The images are so mysterious that I’d have a lot of trouble explaining them, and even if I succeeded they probably wouldn’t help anyone. But they do something good for me. Maybe they engage my visual cortex in the reasoning process somehow. They certainly make these structures feel ‘real’: they’re not longer just symbols on paper.

The other trick was to think more conceptually, more verbally—that is, using words in my mind that express concepts that make sense to me. Again this is a bit hard to explain, because it’s not as if in the old days I was completely unable to reason verbally! But now I’m much better at curating a large collection of verbally expressed insights and using them one after another to solve problems.

So, basically, getting better at math required me to think in more flexible ways. And I think that’s a large part of what math is about!

]]>Thank you, too, for your very interesting perspective. I especially liked the part where you spoke about the need of concrete examples. I think one major, and maybe even the most important, learning step is to make the transition from a lot of concrete examples to the abstract general case. I guess that this is the major division line between scientists and mathematicians. Modern math seems so abstract and general, that I often wonder, how one could even derive such general results, without referring to examples first. I guess, there exists a rare kind of people who really can do it, but the “average” mathematician also needs the more concrete framework to build upon (and tear down again/hide it later :) ).

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