What I think with regards to mathematics is that the axioms are invented, but the truths, e.g. if you assume this you get that, aren’t. But that doesn’t mean an alien will assume the same mathematics as us due to its different development. Likely, an alien won’t even THINK the same way we do, for its brains (or whatever) will have evolved along a fundamentally different evolutionary route. That would certainly frustrate any attempts to “communicate” with aliens.

What do you think is a non-“cowardly” attitude? Can you accept that, say, the Cantor’s infinity does not “exist physically”, yet still consider it a legitimate mental exercise? Or do these people you talk to actually believe the Cantor infinity exists “physically”? What exactly did you say to them? That it’s “just” a mental construct and not a physical reality?

Also, the question of whether or not “ordinary” infinity exists physically is unresolved: in particular, we cannot see whether the universe goes on beyond the cosmic horizon or not. It may or may not, we do not know. Yet, that doesn’t mean I cannot “believe” in such infinity as a useful construct, e.g. I can imagine taking a limit “as x goes to infinity” or “approaches infinitely close to some value”, and the entailed “complete” Cantor-Dedekind real numbers, as a mental construct, and use the mental construct as a model and reason from the model and get useful results, and also find it interesting to study the mathematics of the model itself in its own right.

]]>Secondly, pure mathematicians are very good at evading, precisely the questions you raise about whether certain branches of mathematics are in any sense reality based. Essentially, they believe in their own inventions however fantastic and are outraged if anyone calls them to task. Tell them that Cantor’s Theory of the Transfinite is just a piece of intellectual embroidery and you will meet with a rude reaction. I will never forget the outrage on the face of a pure mathematician when he blurted out, “Good God, this guy does not even believe in ordinary infinity let alone the transfinite!” It was like a Victorian bishop meeting someone who said he had doubts about the divinity of Christ. It has been well said that a pure mathematician is both a Platonist and a Formalist, a Platonist in his study and a Formalist when mingling with the hoi polloi and asked to justify his belief. I view this attitude as cowardly and deceitful, what we call “having it both ways”. Yes, in reply to your question, what I said to ‘these people’ was that Cantor’s theory was simply a mental construct with no more reality or relevance than a pattern in embroidery. But strangely enough Cantor’s definition of ‘cardinal number’ as a set of actual or imaginary objects is very useful and sensible. SH ]]>

Is it “bad” to reduce shapes to number? Not if you can use it to design an airplane engine on a supercomputer. But is it “bad” to say this is the ONLY way and nothing else is useful EVER? Yes. But does anyone actually MAKE that claim? That’s what I’d be curious about. Is there a citation of such a claim?

With regards to diagrams in geometry — for Euclidean plane geometry, to me it seems that they should be maintained because they are very useful for getting the intuition. At the same time, however, the caveats must be mentioned and heeded so one does not draw the *wrong* conclusions from the diagram and one must be able to back up any insights from the diagram with the formal reasoning from axioms and theorems to conclusions (if this is possible at the level of knowledge available) since you are dealing with the idealized world specified by those axioms and also dealing with logic. Euclid, for example, made some assumptions which did not appear in the axioms, but rather were based on experiment with the diagrams. When you get up to the kind of geometry used in, say, General Relativity, however, they become much less useful.

And as for reducing to algebra — it’s good to know that you *can* always reduce to algebra, since that means it is always available to you as a tool, but it doesn’t mean it is necessarily the best way to go about solving a particular problem in actual practice. In theory, you can solve all Euclidean geometry algebraically, but in many cases this can be tedious. Applying the reduction to algebra mechanistically and without thought doesn’t solve problems.

As for this footnote: “I am often tempted to think that civilization (and possibly life itself) is a mistake” — by what condition? In an Atheist universe, life is just a natural process, it is no more a “mistake” than hydrogen fusing to helium in a star. In a Theistic belief system with a God-made universe, life is the creation of God and God is perfection and thus it is not a mistake. Mistake is the domain of humans. Humans made mistakes, neither nature nor God.

With regards to “certainly I do not judge individuals or societies according to their technological and mathematical development. ” Correct. They should instead be judged by how they *utilize* such things. Such things represent capacity and capability for both good and evil. Technological developments, for example, allow for our kids to stand a strong chance at surviving to adulthood, whereas before many would die painfully. This as much as they gave us guns by which kids could be painfully killed by deranged people (although on the whole, in technological societies the net effect is more kids survive now). It’s what we do with what we have that matters.

Also, though maybe it had been developed from militarism doesn’t mean it had to be, or that the alternative is get rid of warworld civilization and go back to dyingkidworld, or that it can only be used for militarism. If present civilization is too militaristic then we need a new form of civilization based on different principles, i.e. a path different from both stay-the-course-with-war-world-civilization and return-to-mass-child-death-world.

]]>You mention that mathematics divorced from reality would be addressed on grounds more like art than on “empirical” grounds, but I’m not sure who exactly considers, say large cardinals in infinite set theory to be “empirical” in the sense of a physical scientists, so the complaint does not make much sense to me.

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