Immanuel Can wrote:
Sure. The laws of mathematics are a set of closed-system-referential, self-coherent abstractions, just as you say. However, they work very well in correspondence to the real world. We see this, for example, in engineering: that a physical structure that is not mathematically sound will not turn out to be functional in the actual world either.
But it will not be sound because of the maths, but because the numbers represent sound structures. The sums are indifferent to what they represent; 2 + 2 = 4 whether the 2s represent steel beams or matchsticks. Or to put it another way, applied maths and pure maths are different realms.
Ah, I see...by "explain" you mean "justify." No, you're right: we cannot "justify" these things, but have to take facts like the basic mathematical or logical coherence as suppositions; but once we do, we find them abundantly confirmed. That first step may not be "justifiable" beforehand, but it is in retrospect...at least, probabilistically.
Of course if we assume various axioms then things will follow from those axioms. But we can do that for anything. If we assume that everything in the world takes place through God's will, then it will follow that everything that happens is an example of God's will. And so for any number of ideas. OK, maths is a lot more useful a theory than the other one, but - strictly speaking - that doesn't confirm it.
But I think the problem I was drawing attention to is not that the basics of logic and maths are assumptions, but that they are different assumptions. They cannot both be confirmed. And also that we do not find the basic assumptions of maths confirmed, on the contrary we find (through maths) they are un-confirmable.
Yes, this is so. All empirical knowing is merely probabilistic, not absolute. But if logic and maths vastly increase our probability of being correct, then how foolish would we be to dismiss such "tools" in favour of, say, random guesses or unexamined tradition.
If empirical knowledge is only probablistic (as you term it), then could do we square it with the notion that logic is reliable because it is drawn from 'reality'? Either logic is validated by reality, or reality is validated by logic, but they can't validate each other, like two drunks holding each other up!
....
But having said all that, we are left with the fact that the empirical world certainly seems to be mathematical, in the sense that the language of maths seems to perfectly describe the relationships within it. Is that because the world somehow
is mathematical, or is it because we have the sort of brains that find mathematical descriptions satisfying? That there might be an alternative way of understanding the world, but we are not wired to comprehend it? It is annoying, but we can never know.
I fear that's where a lot of "Postmodern" talk goes: it throws baby out with bathwater, and loses everything in confusion. Ironically, however, to make its case, Postmodernism itself cannot avoid using logic. For if its conclusions are not in any way verifiable, and have no appeal but to irrational emotions, then there is nothing to make them winsome to any more thoughtful mind.
So I'm careful about claims that "reason isn't reliable," and so forth, whenever I hear people make them...for even if "true" in an absolute sense, they are probabilistically untrue. Reason and logic are themselves reliable processes, and used correctly, lead to vastly more probable results than the alternatives, even taking into account the variability of the empirical world.
Yes, that is obviously right. I'm only making a philosophical point, not a sensible one!
