Can math be ascribed some sort of a physical existence? How could this work, from a philosophical perspective?
I will consider some proposals for doing this and address them each in turn.
Defining Math Programmatically
Is it possible to define math with a software program? Specifically, we note down a specific configuration (programming language, etc.), and we write a program in this language that can accept valid math and reject everything else. Assuming that we wrote all our math papers laying bare all the primitive steps, could we construe this program to be, in some sense, a “definition” of math? Then, could we say that the physical existence of this code on a particular machine lends some physical existence to the concept of math itself?
Well, my first point is that this program can change, depending on whether we want to for example incorporate higher-order logics; indeed, logical formulations of different logics have not been “obvious” in the history of math, so there is no reason to assume that the “universe” of explorable logics cannot expand further in the future. The decision of whether and how to change the program accordingly must then be part of the definition of math, and that is only understood by humans, so the program as written currently cannot be the full answer.
But I also argue that the program can serve as a great practical definition, and can act as a “snapshot” for our current definition of math.
Even still, physical existence is still dependent on the configuration required to run the code, and this configuration ultimately is based in hardware. Hardware can theoretically always be unreliable, and in practice this is actually very much the case, for example in such scenarios as distributed systems. With potentially faulty hardware, it still would require human judgment to decide whether the hardware is indeed faulty and whether the code is running “correctly.” After all, as far as nature is concerned, the electrons just move and do their thing according to the laws of natural science. We impose our wills on our inventions using these laws, and in doing so we impose our standards on what is deemed a “correct” versus an “incorrect” run.
Defining Math Using Language
Alternatively, is it possible to define math as a specific subset of language (a specific set of all valid mathematical statements)? Can we specify the definition of this subset physically? Does language itself have a physical existence?
We can record language in forms such as writings on paper or audio discs. However, ultimately, these physical objects degrade, and the interpretation of the language is still done by us. (Relatedly, speech is generally considered the defining basis for language and linguistics, not writing — humans were speaking before they were writing, thus historically it makes sense to say that writing evolved from speech rather than the other way around.)
Defining Math Using Physical States of Neurons
Can we say that humans’ physical existence, and the physical states of neurons in our brains as we speak, constitutes physical existence of the interpretation of language, and thus what is valid math?
It remains to investigate this idea further.
Parting Thoughts
This question still requires further discussion to come closer to a solution. But generally, it seems that interpretability gets in the way of a physical definition of math — humans would always still need to interpret whether a particular physical construction properly defines math. It would be interesting to see whether the “state of neurons” idea could provide a resolution to this.
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Written December 2020, slightly edited November 2023.
