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I would argue that this is not an assumption. Something exists; we know that something exists; and we know that we know. What existence “is”, what knowledge “is”, how and why knowledge is possible—those are challenging questions. But doubting that anything exists, and doubting that there is any knowledge, seems to require willful negation of fundamental phenomenological facts.
And it’s not far from the existence of knowledge to the existence of “evidence”, since evidence is just, any fact that has implications for the truth; and it is part of the manifest nature of knowledge, that it comes via awareness of facts.
I like Ayn Rand’s related formulation: “Existence is identity, and consciousness is identification”. To be is to be something, and to be aware is to know something.
What do you mean by evidence, in a way that math needs evidence? A person can keep track of mathematical understanding and truth in their own mind, only consulting the outside world (math texts etc.) for inspiration, not evidence. I guess this wouldn’t work if observations are random, so there needs to be expectation that there are math texts out in the world, and finding them is evidence that there is more to find. But a sufficiently smart agent can figure out math all on their own.
I would, in plain language, say that ‘math needs evidence’ is true.
It seems reasonable to think that the study of the natural numbers was the earliest math. I’d imagine that reaching the idea of abstract numbers itself required a lot of evidence.
And mathematical practice since seems to involve a lot of evidence as well. A valid proof seems to exist in the perfect Platonic world of forms and I’m very sympathetic to the sense that we ‘discover’ proofs and aren’t ‘inventing’ them. But finding proofs, or even thinking of searching for proofs seems both necessary in the abstract as well as practically required.
I have been explicitly instructed by math professors to play with new math, e.g. gather evidence of how those systems ‘work’, with the context that doing so was necessary to develop general understanding and intuition of that material.
I meant both empirical and tautological evidence, so general information that indicates whether a belief is more or less valid. When you say that you can keep track of truth, why do you believe you can? What is that truth based on, evidence?
There’s evidence in the form of observations of events outside the cartesian boundary. There’s evidence in internal process of reasoning, whose nature depends on the mind. When doing math, evidence comes up more as a guide to intuition than anything explicitly considered. There are also metamathematical notions of evidence, rendering something evidence-like clear. Hence the question. To figure things out, it’s necessary to be specific. It’s impossible to figure out a large vague idea all at the same time, but some of its particular incarnations might be tractable.
There’s evidence in the form of observations of events outside the cartesian boundary. There’s evidence in internal process of reasoning, whose nature depends on the mind.
My previous comment said:
both empirical and tautological evidence
With “empirical evidence” I meant “evidence in the form of observations of events outside the cartesian boundary” and with “tautological argument” I meant “evidence in internal process of reasoning, whose nature depends on the mind”.
When doing math, evidence comes up more as a guide to intuition than anything explicitly considered. There are also metamathematical notions of evidence, rendering something evidence-like clear.
Yes, but they are both “information that indicates whether a belief is more or less valid”. Mathematical proof is also evidence, so they have the same structure. Do you have a way to ground them? Or if you somehow have a way to ground one form of proof but not the other, could you share just the one? (Since the structure is the same I suspect that the grounding of one could also be applied to the other)
We have two examples of what “evidence” could mean here: mathematical proofs and physical events (things happening in a certain place at a certain time). You can study proofs. And you can study physics. There are hardly any arguments where these two different things are predictably interchangeable, so using the same word for them is a problem. Consider the statement “evidence exists”. Making it specific for our two examples, we get “proofs exist” and “physical events exist”. I’m not aware of a good use for these statements (it’s not at all clear what they could possibly mean).