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Do mathematicians think mathematically?

DuneyBlues

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How does one fully understand a mathematical concept?

By memorizing every formula and plugging it in / by looking for relationships and patterns between different ideas / by having a visual understanding of it ? What what what? I need tips.. different tips from everyone.

Last edited by DuneyBlues on 02 Dec 2011, 5:35 pm, edited 1 time in total.

This post belongs in "Computers, Math, Science, and Technology." That said, the answer is that it depends both on the way one thinks and on what mathematical concept one is trying to understand. Personally, I rely heavily on visual aids, e.g., graphs.

^ I think this question belongs in the philosophy of mathematics.

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There are many perspectives you can take on understanding math, and given the scope and depth of mathematics, there are many perspectives you can form when trying to understand math.

One with I should point out with math is that the different subfields are for a greater part generalizing the math that already exists. It happens in every area of math and also happens in physical and social sciences too.

When you actually get into mathematics and especially in the higher or more general areas, you don't really focus on relationships with numbers: you focus more on structures, decomposition and generalization.

With regards to the analogy of math being like a puzzle, I think that is a valid statement. It takes time for things to "make sense" and to get to that point you often have to practice it for a while and also consistently think about it or to also teach it.

I think the best thing you can do apart from what was mentioned above is to ask "why are they doing this?" and "how is this generalizing some other concept?"

I'll give an example: when you start learning linear algebra and vector spaces you come across inner products and learn about orthogonality. In this kind of context you think about orthogonality as being "vectors" that are independent of each other.

But then you look at things fourier series, functional analysis and hilbert spaces, and then wavelets and you can see that the idea of orthogonality is basically a structured framework for "decomposing macro objects into micro objects": that is, its a structured way to break systems into independent atomic objects.

When you start off learning linear algebra and inner products, this is not so obvious, but as you see things applied to a wide range of phenomena, you begin to see the real concept behind the ideas.

In saying that, if you want to develop those kind of insights, then you need to read a lot of math and think about the questions I stated above. That's how it works in any field whether you are a doctor, teacher, scientist, engineer, whatever: it takes time, effort, and thinking to answer the kind of questions you're asking.

By memorizing every formula and plugging it in / by looking for relationships and patterns between different ideas / by having a visual understanding of it ? What what what? I need tips.. different tips from everyone.

By being able to extend it to any case.

Fundamentally it's about understanding abstractions and knowing how to use logic to show how certain mathematical facts are related. I tend to use visual analogies to picture mathematical objects and concepts in my head.

Personally I get a lot more satisfaction out of deriving formulas than memorizing them. You can't really understand why a formula or theorem is true until you start thinking about how to go about proving it. Sometimes the proofs you get in a typical math text don't give a lot of insight. I prefer having an intuitive motivation before breaking something down into a formal proof.

I usually find myself looking for a way to visualize why a statement might be true by playing a movie in my head or picturing a specific case. A lot of times the general idea for a proof is fairly intuitive even though you wouldn't know it right away by looking at the proof in the textbook. A lot of the mess comes from all the little logical details you have to consider to make the proof air tight. It's also interesting looking at the little exceptions that make all the details necessary. It also helps to go through a lot of examples testing all the assumptions of a theorem and seeing how certain pathological cases make the assumptions necessary.

By memorizing every formula and plugging it in / by looking for relationships and patterns between different ideas / by having a visual understanding of it ? What what what? I need tips.. different tips from everyone.

Hi DunetBlues:

Mathematicians "think" in "Verbal Behavior" as defined by B. F. Skinner in his book "Verbal Behavior".

Developing a repertoire of verbal behaviour for math limited to arithmetic often includes much of everything in daily life because simple "counting" is involved in many basic activities, from buying groceries, to balancing checkbooks, etc.

Carpentry activities are benefited by any math extended to Euclidian geometry and trigonometry, while activities associated with Global travel are benefited by spherical geometry when longer trips are involved, but with shorter trips, Euclidian geometry still works. A simple example is using geodesics to determine the shortest path to make a trip spanning a significant length on the surface of the Earth. With enough experience using geodesics, it tends to get very easy to "think" in terms of spherical geometry by using verbal behaviour specialized more to spheres.

Some branches of math often seem to be nearly useless for practical purposes, but opinions like that are often demonstrated to be wrong. One notable example of this is "number theory" and usages of "Prime Numbers", as in the book "An Incomplete Education", by Jones & Wilson (1987,1995,2006), (2006) page 550 (p. 663?), under "Numbers", section "Prime" for, "Practical Uses" of Prime Numbers: "None whatsoever, unless you find yourself at a mathematicians' convention." Though, now, Prime Numbers & Number Theory are of utmost use and importance in maintaining security using computers and on the internet, and in most all types of basic usages in computer programming building higher-level programing languages.

In terms of "thinking" visually about mathematics, it is moderately easy to visualize higher-dimension manifolds & "objects" in greater than 3-dimensions (some models "hold" that "everything" can be embedded in 11-dimensions, as by that particular model holds "reality" at 10-dimensions). The notational formalism to "do" higher-dimensions is more difficult, but offers guidance on visualizing more details. For example, the old riddle known as "Oblers Paradox" with "background radiation" has an answer in higher-dimensions with "steady-state" models of "island universes". It also helps with practical questions of the physics of stress testing/construction-designs using much the same Tensor calculus.

"Memorizing every formula" tended to get in the way in my studies,as having the verbal behaviour to construct a needed formula from the "basics" offers more conditioning in "thinking" and avoids the excess "luggage". For example, many cumbersome to memorize trigonometric identities have very easy to utilize hints from base-e at powers of imaginary numbers. All of math seems to have these aspects where the "difficult & advanced" parts turn out to be much easier than the alleged "easy" parts.

In hindsight, for me, conic-sections & the quadratic formula were the most important for forming a solid foundation that seems always useful in visualizing advanced problems for the most simple solutions.

Tadzio

I would say learning maths is like building up a model in your head of a landscape that is defined by relationships between things. Like the equations that define motion can look like a flying thing, or the equation that describes a sphere can look like a sphere.

When we understand something in maths its like we recognise the whole object and we feel satisfaction because it is complete.

I think mathematicians think like that. I once worked with 3 of them in a small room and that is the impression I got.

Discovery is guided by a kind of artistic intuition and Justification (i.e. proving theorems) is primarily an exercise in logic, though not completely.

ruveyn

I think a lot of mathematics comes from taking certain concrete ideas that are physically intuitive to most people and going through the process of distilling them to their minimal functional essence. IMO, a lot of the more interesting things in math have come up in this process of digging down and trying to find where the root logical assumptions lie.

For instance, the intuitive concept of an area or volume seems simple enough, but when you break it down and try to define it in the most general way... a way that applies to the amount of "space" taken up by

*any*set in R^n, lot of interesting things happen. It turns out you can generate mathematical sets that are physically paradoxical in terms of having any good way to define the amount of space they take up. It turns out that the existence sets that don't have a Lebesgue measure are a consequence of the

*axiom of choice*which is an assumption that seems intuitive but is independent of any other set theory axioms.

http://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox

So I find it pretty interesting how the world of mathematics has things in it, built on raw abstractions, that seem to defy normal physical and/or geometric intuition. It's like a whole other world.

By memorizing every formula and plugging it in / by looking for relationships and patterns between different ideas / by having a visual understanding of it ? What what what? I need tips.. different tips from everyone.

My undergraduate study of mathematics involved all of the above.

I define mathematics as "the study of patterns." We examine phenomena (whether natural or artificial) and we look for patterns that we can reduce to formulae, or that we can relate to patterns that we have seen and studied before.

Sometimes the memorization of tools is essential. Much of the basic study of calculus, for example, is a question of rote learning of methods of dealing with problems. But later, intermediate study will go back and reexamine the development of these methods by looking at them in the context of their place in the larger pattern of the behaviour of numbers.

Visual understandings can be hugely valuable. When you plot a polynomial equation on the Cartesian plane, it becomes instantly obvious how, for example, the tangent of that function behaves in relation to the function itself. That doesn't tell you instantly what the first derivative of the function is--but it can make the understanding of the derivative easier to access.

But there are other times where visualization is a hindrance. Can we visualize an

*n*-dimensional space? At some point you have to have to have faith in the method of proof and know that the system for representing such a space is 100 percent reliable, even if you cannot visualize it.

_________________

--James

Going off at a slight tangent ( ) I do something similar with programming. I've been a professional programmer for nearly 30 years now and somehow or other I've learned to visualise modules of running code. So for example complex data processing with loops within loops with internal logic sometimes present themselves in my visual mind as coloured flows - like seeing a stream or river, the flows can circulate in loops and vary in width and speed and interact and generate new flows. All this is very abstract and difficult to describe in words.

However what sometimes happens (usually when I'm in bed trying to go to sleep!) I visualise these flows and sometimes there is a break in the smooth flow - a tiny trickle falling off one edge that shouldn't be there - I instantly know there is a bug in a specific part the program code - in this specific example it means a boundary condition is being exceeded, so I have to get out of bed and go fix that part of the program code. My mind doesn't run the code sequentially line by line the same way the computer does - it creates a visual construct of the whole module running. I imagine mathematicians may do something very similar to this with complex formulas?

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*n*-dimensional space? At some point you have to have to have faith in the method of proof and know that the system for representing such a space is 100 percent reliable, even if you cannot visualize it.

I can visualize a method of proof in

*n*-dimensional space by analogy with a lower number of dimensions. The visualization doesn't replace the actual logic, but it can help with motivation.

iamnotaparakeet

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When unleashing the arrow in your avatar you can either think S = UT + 1/2 AT^2 and calculate the appropriate tension required in your bow to give the required acceleration to the arrow taking into account its mass and the duration the bow string will be in contact with the arrow, or you can simply visualise the arc of the flight of the arrow and release it. Each approach has its merits depending on what you want to achieve and both can complement each other.

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