Peter Holmes wrote: ↑Thu May 23, 2024 7:28 am
...that implies that a symbolic expression is
not a complete proposition.
This is not dead wrong, but at least extremely nit-picking. I did not say it's "incomplete," and the idea never crossed my mind. I said that it is not a "proposition"
in the linguistic sense of the term. It's not a sentence, or a declaration about empirical reality. It is in
those definitional senses that it is "not a proposition."
Sheesh.
Look, Pete...if we're going to have a reasonable conversation, a little charitable goodwill is in order, don't you think?
You mix up uses of the words logic and logical. For example, can you see the difference between 'a logical argument', 'a logical rule', 'a logic deals with language', and 'not all uses of language are logical'?
No. It's just the nounal and adjectival uses of exactly the same concept.
Mathematics is a language. And as with any language, its 'application' to reality is not inherent or given in advance. We can and do use it in countless ways, and no one way is 'correct' or 'accurate'. Meaning is use.
Well, meaning is use, but correctness is mathematical. Mathematics belongs to the realm of abstraction, and meaning to the empirical world; but these are not actually two different worlds, as Hegel realized. Rather, one is one kind of representation (the universal) and the other the other kind of representation (the particular) of one-and-the-same world. It would be a sort of naive Platonism to separate them in the way you appear to be trying to do.
To put it simply, maths and logic both have definite applications to the real world, and more precise applications than casual thought can ever achieve. That's what makes them worth having. They work. So there's no strict separation between the abstractions of logic or maths and the particulars of the cases to which they apply.
Thanks. But please read carefully. I didn't mention truth-value. And I explained its irrelevance for validity to you not long ago.
Yes, which makes it all the more startling that you seemed to have reversed yourself here. I would not have expected that error...calling an argument "invalid" when the formal structure is actually valid. If you didn't mean that, then fine.
My point is that deductive validity requires that a conclusion can't contain information not present in the premise or premises of an argument. So mathematical or logical premises - such as rule-assertions - can't entail non-mathematical or non-logical conclusions.
That sounds very much like the naive Platonism I was speaking of. The world of maths and the empirical world are not
two worlds, but
one described obversely. See Hegel.
As I understand it - and I'm not a mathematician - the argument about infinity goes on.
Some arguments about infinity do: this one doesn't. It's too easy to demonstrate. And since actual infinite regresses of prerequisites never start (which maths prove), and cause-and-effect clearly posit a regress of prerequisites, we can know for certain that cause and effect cannot be the product of an infinite regress.
It's inescapably clear. And it's empirically demonstrable, as well, if you simply try to do the calculation yourself. You cannot do it. You'll never start.
There's the unity of the universal and the particular again. The maths say it, and the empirical says it. Pretty conclusive, one must say.