B Is infinity an imaginary number?

[UsableThought]

You need to understand something basic about mathematics. Mathematics is (by and large) about careful definitions, precise axioms and their logical consequences. There are lots of possible sets of definitions and no one true "right" set.

I like the above post, including the qualification I have quoted here. This agrees entirely with what I have been reading lately in a few different books, including the Tim Gowers book I mentioned, https://www.amazon.com/dp/0192853619/?tag=pfamazon01-20. Gowers, by the way, is a http://www.macs.hw.ac.uk/~ndg/fom/gowersqu.html. who seems pretty highly credentialed; all I know about him is that I like his writing.

Anyway if folks can stand another long quote, here is another point Gowers makes, again still early in the above book, about the supposed "reality" of numbers (which is a question this thread seems to have swerved into temporarily). I have bolded a couple of relevant sentences:

A further difficulty with explaining subtraction and division to children is that they
are not always possible. For example, you cannot take ten oranges away from a bowl
of seven, and three children cannot share eleven marbles equally. However, that does
not stop adults subtracting 10 from 7 or dividing 11 by 3, obtaining the answers −3
and 11/3 respectively. The question then arises: do the numbers −3 and 11/3 actually
exist, and if so what are they?

From the abstract point of view, we can deal with these questions as we dealt with
similar questions about zero: by forgetting about them. All we need to know about −3 is
that when you add 3 to it you get 0, and all we need to know about 11/3 is that when
you multiply it by 3 you get 11. Those are the rules, and, in conjunction with earlier rules,
they allow us to do arithmetic in a larger number system. Why should we wish to extend
our number system in this way? Because it gives us a model in which equations like
$x + a = b$ and $ax = b$ can be solved, whatever the values of $a$ and $b$, except that $a$ should
not be 0 in the second equation. To put this another way, it gives us a model where subtraction
and division are always possible, as long as one does not try to divide by 0.​

The last paragraph especially appeals to me. I have read the same sort of thing over & over in books that examine the history of mathematics & its development and various extensions over time. Deemed irrelevant here is the question of whether numbers are somehow "real" in the sense of having representations out in the universe. That question - "does the universe use math"? - is thus a separate one from "what is mathematics?" It's much easier to see how math has evolved & what the game is & how it is played - which is what @jbriggs444 is pointing to. The philosophical debates that arise in math now & then have all seemed to relate to its utility and logical coherence, that is, making sure that further extensions are consistent & usable & don't create backwards contradictions or difficulties. That is very, very different from saying that "numbers are real" in some sense. That's a much trickier question, from what I have read, and there isn't widespread agreement on whether it can be answered, let alone how.

A really short way to put this: The fact we can do math says nothing about whether its objects (e.g. numbers) "actually exist" outside our imagination.

I should add that many folks don't seem interested in ideas such as abstraction, cognition, language, etc. I find them very interesting. So do many, many mathematician/authors. There is Gowers, but also Keith Devlin; Devlin wrote an entire book, The Math Gene, in which he speculated that the human ability to do math may be derived in part from our ability to use ordinary language and to understand relations within large social networks.

 

[PeroK]

I am an engineer, and not a mathematician, so I understand I do not speak in the same precise terms as one. But I struggle to think of an infinity that can't be inverted, and, when it is, would not equal zero. Is there one?

An example where inverting infinities goes wrong is. Let $f(x) = 2x$ and $g(x) = x$, then $f(\infty) = \infty$ and $g(\infty) = \infty$, so:

$\frac{f(\infty)}{g(\infty)} = \frac{\infty}{\infty} = 1$

But

$h(x) = \frac{f(x)}{g(x)} = 2$

So $h(\infty) = 2$

But $h(\infty) = \frac{f(\infty)}{g(\infty)} = \frac{\infty}{\infty} = 1$

So, which is correct? Is $\frac{\infty}{\infty} = 1$ or is $\frac{\infty}{\infty} = 2$?

One of the reasons you need to develop rigorous, well-defined mathematics is to resolve this sort of issue. And, one of the things you have to do is to stop using $\infty$ as a number.

 

[DrClaude]

Invoking the hyperreals in a "B" level thread is the maths equivalent of invoking the stress-energy tensor to explain the SHM of a pendulum!

The hyperreals are at an advanced undergraduate level and depend on a solid grasp of real analysis. They are not suitable for a "B" level thread, IMHO.

I think that this thread has now long been flying over the OP's head. Time to close.