Alec Nevala-Lee

Thoughts on art, creativity, and the writing life.

A dynamic organizing principle

with 4 comments

The mathematical idea acts as a dynamic organizing principle on a certain body of mathematics. It is used as much to generate questions and further ideas as it is to answer questions, that is, to generate theorems. The activity that is generated by the idea is open-ended in the sense that future research is always possible. However, the potential of the idea may be optimized in certain mathematical results…A strong idea may have numerous optimal results associated with it. Moreover, this idea may be joined with other ideas to produce even stronger results. The whole matter is a dynamic flux of mathematical activity with occasional points of relative stability that we call theorems.

The usual sequence of definition, axiom, theorem, proof does not apply to the mathematical idea. The idea may not even precede the result. It is conceivable that the result comes first. In trying to understand the result…we may become aware of what we have called the “idea.” When we have isolated “what is really going on” we would then proceed to apply that principle as widely as our imaginations and knowledge will allow. I conclude that mathematics is not merely a body of facts arranged and justified by a stringent logical structure. The logical structure of mathematics gives theorems their stability, but when we remove logic as the focal point of mathematics and replace it by the “idea” we see that the seeming stability of mathematics is not absolute. The data can be structured in many different ways corresponding to what we are interested in and to the mathematical ideas that arise to do the structuring. These ideas arise in response to the question, “What is going on here?”

William Byers, How Mathematicians Think

Written by nevalalee

September 17, 2017 at 7:30 am

4 Responses

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  1. Add the text represented by your ellipse for me to read please. Maybe it means something.

    Some see fractals. Others feel thermodynamic processes at work.

    Entropy.
    How much high math and logic in an ampersand 🔣 or something like it?

    Imagination made math, creates math and language, under pressure.

    Primacy of storytelling vs the real existential physical world’s concreteness

    I don’t know

    Benj

    September 17, 2017 at 8:20 am

  2. Pyramid schemes began around campfires, not on diner napkins…

    Benj

    September 17, 2017 at 8:21 am

  3. @Benj: Here’s a link to the full text. The passage in question is on page 244 of the book:

    http://aleteya.cs.buap.mx/~jlavalle/filMetInvComp/howMathematiciansThink.pdf

    nevalalee

    September 17, 2017 at 8:38 am

  4. I am still reading this. Thanks.

    Benj

    September 23, 2017 at 7:57 pm


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