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Originally Posted by Disinfo Agent
You're using a notion of "science" that excludes mathematics (let's reserve that term for "pure mathematics", here), because math is not based on physical evidence.
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Right, pure mathematics is a kind of mental game. I do not mean that in any derogatory sense, merely that it is an endeavor that we made up the rules and carry out primarily for our own intellectual satisfaction. That such endeavors may later connect with other familiarities is the point where they make contact with science, not before. I'm just using the standard definition of science.
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Although pure math is not subject to the veredict of physical evidence, I would argue that it is subject to a different standard of proof, which comes from elementary logic. I would also argue, as some authors do, that elementary logic is no less objective than physical evidence.
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But what do you mean by "objective", if not "supported by observations?" If we use elementary logic because it is supported by observations, then we are doing science, because we are making contact with our familiarities. If we are using it just because it is a choice of how we wish to order our thinking, then we are doing pure mathematics. The line is gray because there are combinations, but the dichotomy is useful when the two approaches don't agree (the times we need to use "fuzzy logic", for example).
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In fact, even in the day to day practice of the experimental sciences it's often difficult to disentangle the one from the other, as observations are also to some extent theoretical constructs (you've agreed to this in previous discussions).
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Yes, the two are deeply entangled, but one can still separate them in principle. The difference is essentially in the direction one follows. If one is interested in starting with reality and reasoning back to axioms that represent, unify, or describe some important element of it, while projecting away all that does not fit into the system of validation you are using, then one is doing science. If one starts with the axioms and reasons out the logical equivalences, there is no projecting occuring, and one is doing pure mathematics. The key point is that "success" in the latter endeavor is based in provability and consistency, whereas success in the former endeavor hinges entirely on practical usefulness so inconsistencies (in the complete process, not the "math subroutine") are tolerated constantly. Thomas Hardy's famous toast to pure mathematics is relevant to this distinction.
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In short: I reject the dichotomy between objective natural sciences on one hand, and arbitrary theoretical mathematics on the other. I say that mathematics is every bit as objective as the natural sciences, except that it is grounded on a different kind of evidence (logical, rational, or intuitive).
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But that different grounding is no small detail (just look at how mathematicians test their intuition compared to how scientists do it-- forming proofs versus doing experiments), it is the core of the distinction I'm drawing. Nevertheless, I agree with you that when I refer to its rules as "arbitrary", I just mean that a mathematician can choose different rules and still be doing mathematics (like the various other "schools" of proofs). I don't mean to suggest there is not something special about logic, we certainly learn logic from experience and there is something logical about reality. But many people take that too far and conclude therefore that reality is completely logical, a strange conclusion given that it follows from no valid logical syllogism.
In summary, the distinction is that pure mathematics says "let's choose a form of logic and see where it gets us, clinging rigorously to all avenues where we are led", whereas science says "let's choose a form of logic with a good track record for mimicking reality, confront observations, and not worry terribly about esoteric cracks in the facade if we get the overall picture right". There is certainly a conversation between these approaches which has a lot to do with the logic we normally use in both. I think you are stressing the importance of that conversation, and I don't dispute that, it is very important indeed. But where you see the difference is when you encounter "cracks" in the logic-- inconsistencies don't derail science, but they are a catastrophe for what rests on rigorous provability. The difference is very much the distinction between a "proof" and an "explanation".
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I won't claim that this point of view is better than yours, or that I can prove it is truer, but I put it to you that this philosophy, that regards evidence and logic as the two sources of objectivity, is no less defensible than the philosophy that sees physical evidence as the only possible route to objective knowledge. I put it to you as well that, in any case, the latter philosophy is no more than a philosophy: a good scientist may accept it, but he is not required to.
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I think I see the problem here. I am not saying that the
use of logic is not an inherent part of science, I am saying that the logic is chosen to get
results that connect to observations. The authority in science is that it "checks out" when confronted with observations, while pure mathematics has no need for any checks at all. That means that if the scientist gets the result that works, that informs them that their logic was "good", and they might suggest that the mathematicians use that logic because it seems successful. But the scientist invents a concept that is alien to mathematicians: the crucial concept of "close enough for my purposes".
Pure mathematics could never work that way, it has to already take the logic to be "good", and see where it leads. True, science informs pure mathematics as to useful logical choices, but it leaves a lot of leeway for the fingerprints of the mathematician (constructive proofs, axiom of choice, etc.), and pure mathematics informs the "subroutines" of thinking used by scientists, but it leaves a lot of leeway for the fingerprints of the scientist (boundary conditions, simplifying idealizations, accuracy targets, etc.). So the distinction is important to make, even though one should not oversell it to the point of ignoring the important conversation between them. I think you see me as overstressing the former to the exclusion of the latter, with words like "arbitrary" that could mean "unconstrained", and I agree we should not do that.
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Not every one agrees with that. Many thinkers throughout the ages have argued the opposite, that logic is indeed an innate ability of our species.
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I could think a capability, like the capability of swimming, moreso than an innate ability-- I think it is quite clearly trained, just as magical thinking is trained. No doubt our brains have the capacity for both in various measures, but our training determines which we rely on. Just look at a typical evolution vs. creationist debate. (Or the example I once heard of a young girl whose father told her that he has been Santa Claus all this time, that no real Santa came with toys, and she said "well maybe you brought those presents to
our house because you didn't believe in Santa, but other kids who did believe got theirs from the
real Santa. Imagine losing power over the concept of Santa Claus with your own daughter!) On the other hand, there are situations where logic may let us down, such as when we wish to explore aspects of life that may lead to greater happiness or a richer experience. In those situations, a certain measure of "magical thinking" may be entirely appropriate for bettering our experience (is there not a measure of magic in things like love, art, music, and experience itself, and that all of these concepts can be described completely in terms of logic is itself a form of magical thinking in my view).
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As such, for the purposes of this discussion, I will reject that extension, on the grounds that I have never been shown any conclusive evidence that the whole edifice of logic, from top to bottom, is merely a tool molded by our environment through evolution in a contingent fashion.
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And whether or not that's true doesn't really matter-- we are thinking beings who could choose to go against our evolution and use a different form of logic if it suited our needs better. It is those
needs and the mode we employ to achieve them, not the logic itself, that makes the distinction between science and pure mathematics that I'm referring to.
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As an alternative, I offer a different conjecture: that the rational part of our minds is not a mere product of our environment, but rather the environment itself has been conditioned by the rules of logic, because we live in a logical universe. This is why we've been so successful at using reason to understand the world. It was not natural selection which chose our logic, but logic which chose our universe.
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Certainly all possibilities should be on the table, as each might have its lessons. The question you pose here is probably one of the toughest questions of them all-- do we lay logic over our universe like a template that works for many things, and obliterates what doesn't fit like the way a shadow obliterates the third dimension, or is the universe quintessentially logical and so everything that happens must fit into the "logic template"? I think it is
more logical to adopt the null hypothesis, i.e., the former hypothesis, until one can rule it out with evidence. That doesn't mean we shouldn't entertain all possibilities, it just means that I see flaws in the "positivist" approach that starts from the
assumption of a perfectly logical universe.
Godel's theorem is relevant here as well-- from it we know that there are truths about rich systems like the reals that cannot be proven from a finite set of axioms, and if we allow simply listing all that is true as our set of axioms, then saying "reality is logical" becomes a tautology. We don't know that reality has the richness of the real numbers, but reality conceived of the real numbers, so the concept is part of reality, and entails true concepts that are not logically provable. In short, one cannot equate in all cases truth with provability, unless one defines the axioms of the latter to be the former, which begs the question.