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(1) Let's say I understand how one section of math behaves, maybe the natural numbers, and all of math is connected such that any one section of math can be interchanged for any other (my naive, idealized take on the Langlands Program). If I feel most comfortable in the natural numbers, I can stay there. Like if I feel most comfortable in English rather than Chinese; yet I am not limited in what I can ultimately understand.

(2) Let's also say any mathematical description of material/physical phenomena is just as good as any other for getting the "right answer". Just like I can do classical physics with vectors, algebra, Langrangian, or Hamiltonian, etc.

(3) Mathematical descriptions are semantically neutral, or nearly so, and universally understood how they behave mathematically. Natural language is semantically "loaded". This is why I can just algorithmically carry out algebra on natural numbers without learning new concepts to get the "right answer". A natural language description of some phenomenon may lead to nearly endless debate about what is actually meant/going on. See the century of debate on interpreting quantum mechanics. We can have a mathematical description much before a natural language one.

(4) Even the process of human senses/perceptions + forming concepts is not grounded. It is allowed to change and reorganize. Otherwise how could we ever convince ourselves we were seeing mere shadows in Plato's cave after stepping outside? And a person who is blind can still understand the world. Thus our semantic meanings/understandings likewise are ungrounded and change.

These (1) through (4) may suggest two things:

(a) In that all descriptions, languages, and senses+concepts are interconnected (all are human activities) and ungrounded, are all equally valid means of understanding the world? Valid as in me barking and providing a translation key, and doing all experiments in a manner that different barks pick out different empirical results, are equally valid as a more traditional manner.

(b) What evidence suggests we are doing more than just endlessly carefully reorganizing different "languages"? I can come up with we have more mathematical descriptions and scientific theories than the past, but what suggests this will end? A scientific theory saying so? But how does that square with (a)? How does barking suggest anything other than being a way to organize?

(This is all for curiosity's sake, please ask for clarifications too.)

J Kusin
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  • Nice question, I think Wittgenstein, claimed that it is possible all debate is about using different terminology and if we agree we find out, there is no practical difference – Nikos M. Nov 03 '21 at 17:35
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    Also in a sense, constructivism in pedagogy, claims just that, one can create equally valid descriptions of the world as the other, all though different – Nikos M. Nov 03 '21 at 17:38
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    This is a defense of the opposite view: https://ndpr.nd.edu/reviews/fear-of-knowledge-against-relativism-and-constructivism/ – Nikos M. Nov 03 '21 at 17:39
  • @NikosM. Thanks for the links and terms, that was part of my hope posting, to get someone more knowledgeable who knows where to explore these ideas. – J Kusin Nov 03 '21 at 17:49
  • Mathematics is not semantically neutral, uninterpreted theorems are just meaningless strings of symbols. Mathematical fields are not interchangable either, even when formally equivalent, semantic information that drives them is disregarded in the equivalence, and it is connecting semantically distinct domains that makes the equivalence fruitful. Allowing change and reorganization by itself does not entail lack of grounding, only fallibility. "*Facts are hard things which do not consist in my thinking so and so... they are bound to press upon me at last*", Peirce. As in getting hit by a rock. – Conifold Nov 04 '21 at 00:36
  • @Conifold And have we ever come across a fact? Wouldn't an atomist, structural realist, or radical platonic realist like Tegmark have something else to say about this "rock"? – J Kusin Nov 04 '21 at 02:06
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    Sure. The facts are not in what realists and the rest argue over, they are in what happens to them. This line of reasoning is akin to the fallacy of the heap: we cannot describe precisely what a heap is, therefore there are no heaps. – Conifold Nov 04 '21 at 03:25
  • @Conifold I don’t want to deny heaps and rocks entirely with the above. Rather let the latest scientists have their quantum field conception of it, the layman theirs, Tegmark his, and the blind and deaf person theirs. If there’s a dog outside, I can know it by seeing, by it hearing it, by smelling it, or even by hearing my own dog bark at it. I could even devise a Rube Goldbergian detector to learn its presence by orienting my body by proprioception. – J Kusin Nov 04 '21 at 05:03
  • @Conifold, having semantics, a mathematical field can still be expressed through another field, either with same semantics, or a translation of semantics. This is not a no-go theorem. – Nikos M. Nov 04 '21 at 09:58
  • Relativism is a complicated topic... regarding philosophy of science we can consider [Feyerabend's "anything goes"](https://plato.stanford.edu/entries/feyerabend/). Having said that, following your example, when you "bark", someone (dogs included) understand you? If not, maybe it is not so easy to assert that "anything goes". – Mauro ALLEGRANZA Nov 04 '21 at 11:17
  • "we have more mathematical descriptions and scientific theories than the past..." Yes, and they are much better than that of the past; similar to "barking": do you think it is possible to use Aristotle's law of motion to compute the trajectory of a rocket landing on the Moon? to use Galen's medicine to design anti-COVID vaccine? – Mauro ALLEGRANZA Nov 04 '21 at 11:28
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    @Mauro ALLEGRANZA No we wouldn’t use such tools. The structure of their content doesn’t match with goals of landing on the moon. What I am saying is the rocket could be programmed by a person moving grains of sand in a structured manner, or another literally barking with encoded meaning. Or the manual to build it given by pictures and no words or equations, or by a many-sided solid a blind person can feel about. – J Kusin Nov 04 '21 at 15:58
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    @MauroALLEGRANZA there is no a-priori reason one cannot produce an effective medicine for COVID using a completely different description of the world than molecular biology – Nikos M. Nov 04 '21 at 16:53

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