Is Abstract the Best Way to Describe Higher Math?

[johnqwertyful]

Has anyone else found that this is not the best way of describing higher math? I feel like abstract is not a good categorization of higher maths. I've found the hard parts, what I spend most my time on and what's the hardest to keep track of are technical things and subtle details. Random variables converge in many different ways, probability, distribution, mean, almost surely, etc. Some modes of convergence imply other modes, and under certain conditions those other modes imply the first modes, but not always. I wouldn't really call it abstract, just technical and detailed.

I'm not saying there isn't abstraction, because there is, it's just that it's not the first word I would use. Would you call higher math "abstract"? Is there a better word?

 

[Simon Bridge]

It depends what you call "higher math".
The statement could be taken as a definition.

i.e. the validity of the statement depends on it's context.

 

[Stephen Tashi]

Random variables converge in many different ways, probability, distribution, mean, almost surely, etc. Some modes of convergence imply other modes, and under certain conditions those other modes imply the first modes, but not always. I wouldn't really call it abstract, just technical and detailed.

Pure mathematics is "legalistic". It's best to admit this even though legalism, law and lawyers are disdained by many people who pursue science and mathematics. One must read the fine print in mathematical definitions. Definitions mean what they say. Putting them in one's "own words" or replacing them with one's own intuitive ideas isn't a reliable way of proceeding.

 

[disregardthat]

A lot of "higher math" is heavily based upon categorical terms, which is a very abstract setting. Often you will see definitions, theorems and proofs using very general categorical constructions, which I would say is an argument of itself as being "abstract" in some sense.

In my opinion, what characterizes abstraction is the ability to generalize. If you work abstractly, you will see the possible generalizations more clearly.

 

[jedishrfu]

Mathematicians look for the hidden pattern behind things. They abstract it to more general principles. As an example, we look at rules for doing arithmetic as simply rules to follow whereas a mathematician will wonder what if I change this rule or what if I throw it out or what if I add an additional constraint.

This search for patterns has led from natural numbers, to integers, to reals, to complex, and beyond to quaternions, vectors, tensors and differential forms and the abstraction continues...

 

[HallsofIvy]

I would agree that it is not reasonable to say that "higher mathematics" is "abstract" because all mathematics is "abstract". Even when we add 1+ 1 we are abstracting from "one cow", "one turnip", or even "one object" to just "one".

 

[johnqwertyful]

Pure mathematics is "legalistic". It's best to admit this even though legalism, law and lawyers are disdained by many people who pursue science and mathematics. One must read the fine print in mathematical definitions. Definitions mean what they say. Putting them in one's "own words" or replacing them with one's own intuitive ideas isn't a reliable way of proceeding.

This has been my experience. A lot of math is finding some weird fringe counterexample to intuition. I find that legalistic has described my experience much better than abstract, although there is abstraction I find the legalism comes to mind first.

 

[johnqwertyful]

I would agree that it is not reasonable to say that "higher mathematics" is "abstract" because all mathematics is "abstract". Even when we add 1+ 1 we are abstracting from "one cow", "one turnip", or even "one object" to just "one".

I guess this is true, but so little of my time studying is spend on trying to abstract in this sense. The majority of my time is trying to prove things using the precise formulation of Peano arithmetic, for your example. I guess to say my thoughts more precisely, the legalism of mathematics is takes up much more of my thoughts than the abstraction.

 

[jedishrfu]

This has been my experience. A lot of math is finding some weird fringe counterexample to intuition. I find that legalistic has described my experience much better than abstract, although there is abstraction I find the legalism comes to mind first.

I would agree with this sentiment. My son is studying law and was told by a prof that the best case study writers were people who were math undergrads or higher. They paid attention to detail and the fixed definitions of legal terms.

 

[homeomorphic]

Putting them in one's "own words" or replacing them with one's own intuitive ideas isn't a reliable way of proceeding.

I find this a problematic way of putting it. Putting things in your own words and intuitive ideas IS a reliable and even necessary way of proceeding, although there can be such a thing as bad intuition and bad ways of putting it in your own words. It's just that intuition isn't the FINAL way of proceeding. It's a perfectly good initial way of proceeding, up until the very last bit of icing on the cake when you try to make it all precise and write your proofs. Who says that just because you try to be formally correct, you haven't made any mistakes in doing so? Mistakes are much easier to spot when you have the intuition, as well as the logic.

It's just that in the end, you, ideally, want to dot your i's and cross your t's and get everything right logically, according to the exact definitions. I say ideally because, in practice, we know mathematicians, outside basic undergraduate and graduate level math classes, don't actually spell everything out 100% rigorously in many cases because if they did, it would just take too long to get anything done. And even then, physicists still tend to move a lot faster, and in some cases, get more done, because they are even less rigorous.

I find that when you find counter-examples to intuition, often, that just means you need better, more precise intuition, not that the intuition is just getting destroyed and that's it.

Whether math is abstract or not depends a bit on how you do it.

 

[Stephen Tashi]

Putting things in your own words and intuitive ideas IS a reliable and even necessary way of proceeding, although there can be such a thing as bad intuition and bad ways of putting it in your own words.

It is a reliable method of proceeding only for people who have considerable experience in dealing with mathematical legalism. It depends on being able to formulate statements with great precison, being able to understand and state things in terms of logical quantifiers "for each" and "there exists", paying attention to distinction between phrases like "there are two..." and "there are exactly two...", etc. If you take a legalistic mathematical definition and put it in your own words correctly, it comes out to be another legalistic definition.

So, I think the best advice for most predominately-questioners in the math section is that they not put mathematical definitions in their own words. Most of the predominately-answerers know how to rephrase definitions in their own words without getting into trouble.

Intuition is indespensible, but it takes a very sophisticated kind of inutition to make progress in advanced math. I suppose that when answering a question on the forum, we can attempt to fix a questioners imprecise statements and bad intuition in two basic ways - namely 1) clarifty the language or 2) fix the intuition by giving better intuition. I tend to emphasize fixing the language because the question itself is often unclear until the language is fixed.

There are advantages and definite limitations to approaching mathematics and science as "right talking" - i.e. speaking correctly. I've met people who do well in some technical subjects but can't for the life of them speak or write in a precise manner.

 

[homeomorphic]

It is a reliable method of proceeding only for people who have considerable experience in dealing with mathematical legalism. It depends on being able to formulate statements with great precison, being able to understand and state things in terms of logical quantifiers "for each" and "there exists", paying attention to distinction between phrases like "there are two..." and "there are exactly two...", etc.

For me, I think intuition was key to my success right from the beginning of real analysis class. Before then, I did take a class that was sort of an intro to proofs/naive set theory that played the role of focusing on the logic, but that was all the preparation I needed. Being too focused on that stuff in real analysis would not have been necessary. So, here, rather than considerable experience, I'm talking about the 2nd proofs class I took, in which intuition was one of the keys, although writing out complete logical arguments did play a role. I think you never should completely turn your back on intuition, although you probably need to focus a bit more on logic at a certain stage.

If you take a legalistic mathematical definition and put it in your own words correctly, it comes out to be another legalistic definition.

That's pretty much true, but it's usually helpful to have a kind of mental short-hand. The definitions are much more memorable and useful that way. For example, when I first saw epsilons and deltas in calculus, I considered it useless back then, and I still consider it useless in the form it was presented to me in because I didn't have the picture in my mind of what it meant and what the point was. I don't think there's much value in studying it without the picture in mind. It just becomes an empty exercise in torturing the students.

 

[Stephen Tashi]

It just becomes an empty exercise in torturing the students.

Some part of education must involve having the students adapt their minds to mathematical concepts - rather than adapting mathematical concepts to fit those already in their mnd. The typical intuitive presentation of an epsilon-delta definition involves some idea of "approaching". It invites a student to think of some process that would take place in time. The actual epsilon-delta definitions don't refer to any dynamic process. The idea of a limit as involving a dynamic process is an valuable intuition, but so is the more sophisticated intuition that the use of logical quantifiers can capture an essential aspect of a dynamic process without using any dynamics.

Another pitfall of approaching mathematical definitions as ordinary language is that students think that the definitions define single words or phrases like "limit", "compact", "confidence",. Mathematical definitions only give logical equivalences between two statements. Individual words in the statements ( such as "limit" or "approaches" in the definition of the limit of function) aren't defined by the definition itself. Students in elementary school are taught that to understand a sentence, you should break it into its invidual parts and analyze each word. This won't work in formal mathematics.

So I don't think its wise for most students to follow the standard advice one uses to study many liberal arts subjects - namely 1) Rephrase things in your own words 2) Analyze the meaning of each phrase in a statement separately. It's best not do either of those except as a last resort.

 

[homeomorphic]

Some part of education must involve having the students adapt their minds to mathematical concepts - rather than adapting mathematical concepts to fit those already in their mnd. The typical intuitive presentation of an epsilon-delta definition involves some idea of "approaching". It invites a student to think of some process that would take place in time. The actual epsilon-delta definitions don't refer to any dynamic process. The idea of a limit as involving a dynamic process is an valuable intuition, but so is the more sophisticated intuition that the use of logical quantifiers can capture an essential aspect of a dynamic process without using any dynamics.

When I first encountered it, it was presented almost as if it was just fiddling with inequalities for no reason. The intuition I'm talking about (that is the intuition for the epsilon-delta definition, not just the intuitive concept of limits in terms of what things are approaching), which most calculus students lack, is mostly being able to interpret absolute value in terms of distance, and draw a horizontal strip representing epsilon and a vertical one, representing delta. In Stewart's book, he does try to get that picture across somewhere, at least in the exercises. It seemed to me when I taught recitations, that it was mostly lost on the students, anyway. I'm not sure they profited from it. When I explained it to them in a tutoring setting, I think they were able to at least get something out of it, although it was difficult to get the whole thing across to them, and I'm not sure how well I succeeded.

So I don't think its wise for most students to follow the standard advice one uses to study many liberal arts subjects - namely 1) Rephrase things in your own words 2) Analyze the meaning of each phrase in a statement separately. It's best not do either of those except as a last resort.

I would have to disagree. Many books even recommend trying to summarize proofs. It's kind of like telling your soldiers not to bring their weapons to battle because you are afraid they are going to hurt themselves. Well, what's the point of them going to battle then, if they are just there to get slaughtered? Better to let them make mistakes and try to correct them.

 

[1MileCrash]

I think all of math is "abstract", but one doesn't realize it until higher math.