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I am curiosity about if thought experiments have importance in mathematics.

I know about famous thought experiments in physics and also some example in mathematics. Professors who followed that mental image obtained a great theory or theorem, or advance in their research, and seem as direct consequence of such mental image (what they caused) or thought experiment.

In physics I believe that these thought experiments are more popular and well known, but what about the importance of thought experiments in mathematics? In mathematics, the more important thing is rigor or examples that you can write on a sheet of paper. But is that all?

Question. What is the importance of thought experiments in research in mathematics? Have these any importance? Refers the literature if you need it, and I try to find and read these information. Many thanks.

I am asking about the importance of thought experiments that eventually can provide to a professor researching in some field a great advantage or a privileged point of view in his/her work. I am not asking about ingenious or artificious ideas, just about simple thought experiments that provide to the proffesor the idea for the solution of a remarkable theorem.

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closed as primarily opinion-based by Shaun, Rohan, Lord Shark the Unknown, user21820, Ennar Dec 20 '17 at 15:38

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ I am not sure what a thought experiment even means for math. One does mathematics by thinking. The closest thing to an experiment in math is a computation of special cases (by hand or by computer). What would it mean to do such a computation by pure thought is utterly unclear to me. $\endgroup$ – Moishe Kohan Dec 20 '17 at 13:30
  • $\begingroup$ Possibly the analog in mathematics for the notion of a thought experiment in physics is performing various estimation and safety checks, such as seeing if the form of the answer makes sense when large or near zero numbers are used, investigating simple special cases, seeing what happens when non-dominate terms are dropped, etc. $\endgroup$ – Dave L. Renfro Dec 20 '17 at 17:09
  • $\begingroup$ Here there is Wikipedia dedicated to Thought experiment. That I meant if I am right was an example from Hyperspace: A Scientific Odyssey through Parallel Universes, Time Warps, and the Tenth Dimension, by Michio Kaku, Oxford University Press (1994). If I remember well in the third chapter. You can find the book in the library of your university or city since it is very popular. Many thanks @MoisheCohen $\endgroup$ – user243301 Dec 20 '17 at 21:04
  • $\begingroup$ Here there is Wikipedia dedicated to Thought experiment. That I meant if I am right was an example from Hyperspace: A Scientific Odyssey through Parallel Universes, Time Warps, and the Tenth Dimension, by Michio Kaku, Oxford University Press (1994). If I remember well in the third chapter. You can find the book in the library of your university or city since it is very popular. Many thanks @DaveL.Renfro $\endgroup$ – user243301 Dec 20 '17 at 21:04
  • $\begingroup$ Good morning @reuns you've here the Wikipedia's article dedicated to Thought experiment. My question isn't about experiments with a computer. I'm interested in science as is shown in my Question. Also in different details as the so called serendipity, you can find it the definition 1 from this Wiktionary. I provided the source from what I interpret a paragraph as a thought experiment. My question isn't about a programming language ... is much more important than it. $\endgroup$ – user243301 Dec 21 '17 at 9:10
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I'm of the opinion that all of mathematics is a thought experiment. Unlike the empirical sciences which rely on observations mathematics is a creative discipline which only requires your imagination. Axioms are the beginning point of the experiment and theorems the result. For example, proof by contradiction makes this transparent as it uses thought experiment style reasoning to show something is impossible by imagining it was possible and deriving an absurdity.

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  • $\begingroup$ Thank you very much for your answer, I will not accept any answer because I respect for all those who participated and since the question was put on hold, but I have already read your answer once and will do so in the next few hours. It is very valuable. $\endgroup$ – user243301 Dec 20 '17 at 20:46
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I guess the answer (as everything in math) depends upon definitions of what "thought" and "experiment" is. When you're dealing, for example, with infinity... is that an instance of "thought experiment"? In case it is, the answer is obvious: math is built up from such thought experiments taking their place literally at each and every tiny place. In case it is not, then the essence of "thought experiment" remains to be an open question...

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  • $\begingroup$ Thank you very much for your answer, I will not accept any answer because I respect for all those who participated and since the question was put on hold, but I have already read your answer once and will do so in the next few hours. It is very valuable. $\endgroup$ – user243301 Dec 20 '17 at 20:46
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If I understand you correctly, you want to know how thought experiments, i.e. being aware of one's conscious experience, can guide research in mathematics. Interpreted this way, I would refer you to Mach's work, but also Husserl's phenomenology. Before dedicating himself fully to philosophy, Husserl obtained a Ph.D. in mathematics. In his first book, The Philosophy of Arithmetic, he aims to synthesize the psychology of impressions with the concept of number.

I don't know whether you'll find an algorithm for designing thought experiments there, but some awareness of what goes on in the mind when we assign numbers to things, or investigate their structure, might inspire you to arrive at your own.

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  • $\begingroup$ Thank you very much for your answer, I will not accept any answer because I respect for all those who participated and since the question was put on hold, but I have already read your answer once and will do so in the next few hours. It is very valuable. $\endgroup$ – user243301 Dec 20 '17 at 20:46