Mathematics

Having an almost three year old daughter leads me into deep philosophical questions about mathematics. :-) Really, I am concerned about the concept of "being able to count". People ask me if my daughter can count and I can't avoid giving long answers people were not expecting. Firstly, my daughter is very good in "how many" questions when the things to count are one, two or three, and sometimes gives that kind of information without being asked. But she doesn't really count them, she just "sees" that there are three, two or one of these things and she tells it. Once in a while she does the same in relation to four things, but that's rare. Secondly, she can reproduce the series of the names of numbers from 1 to 12. (Then she jumps to the word for "fourteen" in our language, and that's it.) But I don't think she can count to 12. Thirdly, she is usually very exact in counting to four, five or six, but she makes some surprising mistakes. Yesterday, she was counting the legs of a (plastic) donkey (in natural size), and she had to move around to see all of them: she managed to come to the conclusion that the donkey had six legs. Fourthly, she usually forgets one of the things or counts one of them twice when she is counting to seven, eight or nine. Finally, she never asked her parents what is the number "next" to some other number (say, the numbem "next" to twelve). Now, do you think that she can count? And to how many things can she count?

Typical statements (first order) of the Peano Axioms puzzle me. Neither a mathematician nor logician, I find myself thinking the following: One would hope that arithmetic is consistent with the world as it is. So the axioms of arithmetic should be true in a domain containing the items that populate reality, e.g., a domain containing this keyboard upon which I now type. But this keyboard is neither identical to zero nor is it the successor (or predecessor) of any whole non-negative number. So what's with, e.g., (Ax)((x = 0 v (Ey)(x = Sy))? On what would think its intended interpretation, the axiom (theorem in some versions) seems false "of reality." And some other typical items of (first order) expositions seem either false or at least meaningless, e.g., (Ax)(Ay)(x + Sy = S(x + y)). What could be meant by "the sum of this keyboard and the successor of 6 is equal to the successor of the sum of this keyboard and the positive integer 6? Unless one has already limited the domain to exclude typical non-arithmetic items, then stating the (first order) Peano Axioms with leading universal quantifiers seems to produce false and false or meaningless statements. So how would one try to change/complicate the (first order) axioms to avoid this? I recall reading somewhere that in some of his work Tarski would use a predicate for non-negative integers to limit the scope, something like "for all x, if x is a member of the non-negative numbers, then...." But how else might I think about this? Thanks for helping un-confuse me. Or don't we care if the Peano Axioms are not true of the world we live in? Wayne W.

For a long time I have been very concerned with clarifying mathematics, primarily for myself but also because I plan to teach. After decades of reading and questioning and thinking, it seems to me that the philosophical views of mathematics are nonsensical. What does it MEAN to question whether mathematical objects exist outside of our minds? It sounds absurd. It seems clear to me that mathematics is a science like all the others except that verification (confirmation) is different. It is the science of QUANTITY and its amazing developments and offshoots (like set theory). And all sciences are products of our minds. They are our constructions, as are most of the physical objects in our immediate worlds. Shoes, sinks, forks, radios, computers, computer programs, eyeglasses, cars, planes, airports, buildings, roads, and on ad nauseam, are ALL our constructions. Nature didn't produce any of them. We did. What does it MEAN to speak of a "PHYSICAL" circle? A circle is OUR IDEA of a plane locus equidistant from a point. A transistor is no less real because it is OUR invention. How can anyone MAKE such a distinction? Who cares what Plato thought about mathematics? He didn't know what an algebraic number is. He didn't know what a p-adic number is. Hardly any mathematics had been invented yet twenty five hundred years ago. Why do people respect in his speculations, his fictions? And the same is true of the other contenders that presume to account for mathematics. We are surrounded by our inventions and their properties. My father used to have to get his car greased. No one does that anymore. Now we have much better bearings. Can you please explain to me why there is so much bizarre speculation about the nature of mathematics? I hope you answer. I am truly perplexed. (I started as a philosophy major but switched to electrical engineering.) Thank you, George F.

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