Are there better concepts than number for math?

[AnEboss]

People are limited, true enough, but we needn't be people forever. The reality is that we seem to be obsessed with this idea of progress as a exponential curve fast approaching a asymptote of some kind. Examples include the singularity, collective consciousness, or Armageddon. I find these ideas unconvincing because it's not how human history looks to me. There are times in history of rapid advancement and changes in civilization, there are long historic plateaus, and there are periods of outright decline and chaos. We have been in a period of radical change for so long that it seems that it must continue forever, accelerating, but this is unsustainable.

When I was a kid I asked my dad a question "Will runners eventually break the sound barrier?" My thinking was that if each runner was breaking records of the previous runners, there would eventually get to be a runner fast enough that he can outrun sound-which I already knew was clearly limited in velocity. The point is that situations change.

We may reach the end of our capabilities for a time, but then we will regain traction in a new way. Things don't spiral wildly towards an asymptote only to stop suddenly. They will stabilize, build up tendency, and start again. If there is a better math out there, we will eventually understand it. Unless we all die horribly and suddenly in a gamma burst or nuclear fallout. No guarantees and all that...

[Philosophy Explorer]

Used to think that, with respect to math and science, that every term needed to be defined and every theorem needed to be proven. Then I learned in these areas, that just isn't so. For example, in math, we learn you can set out in algebra with a set of assumptions called axioms that are used in proving basic theorems. And we start out with undefined terms in math such as number e.g. As I pointed out, by fixing math with defined terms, you may never discover new areas we may call math (and the same may hold for the sciences). Again, with respect to math, the challenge may be to determine what its extent is, what new discoveries await us. And the same may hold for the sciences.

PhilX

[AnEboss]

I agree with you. Do you have any ideas in particular about what those discoveries may be? I'd be interested to hear what has your wheels turning in more specific terms.

[Philosophy Explorer]

I agree with you. Do you have any ideas in particular about what those discoveries may be? I'd be interested to hear what has your wheels turning in more specific terms.

Actually no, but sometimes math is behind the sciences and sometimes it's ahead. Currently there is a great deal of interest in consciousness as witnessed by this forum. Often math is wedded to different branches of science (I think one of the primary ones is still physics); if it becomes intertwined with consciousness, there may be new math (one more interesting note is different parts of math help each other out, e.g. completing the square helps out in solving integration problems.

PhilX

[Alan Masterman]

Maybe I missed something, but why do we need a better concept than number?

Yes you did miss something.

PhilX

Thanks for your in-depth critical analysis, Philx. Please bear with me, but would you mind explaining what is being proposed as substitute for a theory of number, and what special problems it will solve which currently cannot be solved by number theory? I am not a professional mathematician, you understand, and I'm certainly not up-to-date with the latest developments, but the hypothesis that mathematics can be carried on without a theory of number is one that I have not come across. I presume you would argue that the researches of Peano, Riemann, Frege, Russell et al are irrelevant to modern mathematics - so what is the magic bullet which will replace number theory?

[Philosophy Explorer]

In response to A. Masterman, one possibility is nonstandard analysis as was headed up Abraham Robinson (now deceased). Yet another possibility is Leonard Euler's theorems, one of which was recently used to prove something in relationship to string theory. Then we also have unexplored territory in set theory.

I've read that over 200,000 new theorems are published each year to give you an idea as to the scope of math. Being that math is undefined doesn't seem to be a hindrance towards its expansion and I can argue just the opposite.

I would recommend getting ahold of the book The Mathematical Experience by Davis and Hersh. Although copyrighted in 1981, you should find it stimulating as it covers a lot of ground.

PhilX

[Reactor]

Here's what I'm driving at. Numbers started off in history to denote quantity and for counting (don't know which occurred first). Then the zero concept was invented. Next we get negative numbers. Then complex numbers. Then we extend other things based on number.

For example the factorial. It was defined based on positive integers. Next it was defined on zero. Now it's defined on the entire real number line except for the negative integers.

Is there a better concept that would take into account all of these different situations, and more?

PhilX

Numbers are just one of many sets (or classes) of symbols and concepts we use to work in mathematics.

Numbers have their place in arithmetic and to express quantity in other mathematical subdivisions. Letters serve in algebra, lines in plane geometry, and signs have many uses. The sum of all of our mathematical symbols and the concepts for which they are "shorthand" notation keeps growing with the invention of new systems of mathematical thought.

Since the choice of particular symbols arise during the formative years in any discipline, I seriously doubt that any future "grand symbolic scheme" will replace those favored symbols.