# Is there a role of mathematics in the development of human consciousness?

interesting question. not sure I understand it exactly. but I can refer you to some fascinating work that touches on it -- mostly anything by Douglas Hofstadter, but you might start with Strange Loops and/or Godel Escher Bach ... both spectacular works that trace the essence of consciousness to self-referential recursive (mathematical) processes ... he'd be a good place to start. best, Andrew Pessin

# One of the obvious ways computers are limited is in their representation of numbers. Since computers represent numbers as bit strings of finite length, they can only represent finitely many, and to a finite degree of precision. Is it a mistake to think the humans, unlike computers, can represent infinitely many numbers with arbitrary precision? We obviously talk about things like the set of all real numbers; and we make use of symbols, like the letter pi, which purport to represent certain irrational numbers exactly. But then I'm not sure whether things like this really do show that we can represent numbers in a way that is fundamentally beyond computers.

This one is basically above my pay grade, but I'll take a stab. I share your doubt that humans "can represent infinitely many numbers with arbitrary precision" in any way beyond what we find with computers. After all, our own hardware (our brain) is finite in the same ways/senses as are computers, so if sheer finitude establishes the limits of representation it's hard to see why we would differ from computers. If, on the other hand, you're imagining this as an argument for dualism -- i.e. our minds are distinct from our brains because they have infinite capacity in a way that our brains don't -- then you would definitely first have to prove the infinite capacity of our minds. Simply writing or thinking "pi" isn't enough; the fact that "pi" represents something infinitely expandable/expanded doesn't make the symbol "pi" infinite. The clearest proof would be if we could grasp (say) the complete infinite expansion of pi in one mental glance -- but we can't. At best we can grasp THAT the expansion goes on...

# 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...

Rather than answer I will merely invoke a classic Sesame Street episode. Grover is counting oranges: one, two, three etc. And again: one, two, three. Then someone else comes in with a basket of apples and asks him to count these as wells. But he breaks into tears. Alas, he can count oranges but he has never learned to count apples. ap

# 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?

Apparently you have, if you wrote this question! :-) (People also like to talk about the immense number of neural connections within our brains -- I don't know how that number compares to the ones you mentioned, but I believe it's pretty brain-melting too!) ap

# I have been reading discussions on this site about the Principia and about Godel's incompleteness theorem. I would really like to understand what you guys are talking about; it seems endlessly fascinating. I was an English/history major, though, and avoided math whenever I could. Consequently I have never even taken a semester of calculus. The good news (from my perspective) is that I have nothing to do for the rest of my life except for working toward the fulfillment of this one goal I have: to plow through the literature of the Frankfurt School and make sense of it all. Understanding the methods and arguments of logicians would seem to provide a strong context for the worldview that inspired Horkheimer, Fromm, et al. So yeah, where should I start? Do I need to get a book on the fundamentals of arithmetic? Algebra? Geometry? Or do books on elementary logic do a good job explaining the mathematics necessary for understanding the material? As I said, I'm not looking for a quick solution. I...

lucky you, with so much time on your hands and with such interesting interests! there are numerous secondary expositions of Godel etc. -- I personally love Douglas Hofstadter's way of explaining it (in Godel Escher Bach and also his more recent Strange Loops) ... but Rebecca Goldstein has a recent book on it (haven't read it, can't speak to its quality) -- http://www.rebeccagoldstein.com/books/incompleteness/index.html good luck ap