I've recently read that some mathematician's believe that there are "no necessary truths" in mathematics. Is this true? And if it is, what implications would it have on deductive logic, it being the case that deductive logical forms depend on mathematical arguments to some degree.
Would in this case, mathematical truths be "contingently-necessary"?
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...
We use logic to structure the system of mathematics. Lord Russell was described as bewildered upon learning that original premises must be accepted on some human's "say so". Since human knowledge is so fragile (it cannot have all conclusions backed up by premises), is the final justification "It works, based on axioms accepted on faith"? In short, where do you recommend that "evidence for evidence" might be found, if such exists in the anterior phases of syllogistic construction. Somewhere I have read (if I can rely upon what little recall I still have) Lord Russell, even to the end, did not desire to rely on inductive reasoning to advance knowledge, preferring to rely on deductive reasoning. Thanks. Your individual and panel contributions make our world better.
I've read in several places that scientists have estimated the number of atoms in our galaxy to be (very) roughly 10 to the 65th power. This is an extraordinarily huge and basically incomprehensible number. However, this figure is more than 100 times smaller than the number of ways I could arrange the ordinary deck of playing cards I have in my hands. [52 factorial is approximately 8 x 10 to the 67th power].
Pardon the exaggeration, but how can I keep facts like this from melting my brain?
I know some philosophers think numbers exist, and some others think the opposite. Do some of you think that this question is or may be "undecidable"? I mean, perhaps both the idea that numbers exist and the idea that numbers don't exist are consistent with all other things that we believe (do not contradict any one of them). Do you think this might be right?
First, is it true that academic philosophers reside in ivory towers? And that their ivory tower is filled with books and greek sculptures?
Second, There seems to be an interesting feature of many logicians or philosophers of language, that they have a background in the field of mathematics or being related to the field of mathematics in some other way. Is this in your opinion a coincidence? Does the field of mathematics grant those capable of handling it some clarity of mind or perspective in observing the world? This could be interpreted as a question to what sort of intelligence, if any, is more favorable to logicians and philosophers of language(presupposing that the distinctions made in the theory of multiple intelligences hold).
It was an interesting and, in my opinion, true prediction of Alfred N. Whitehead when he said that science in its evolution becomes more and more mathematized.