## Negative numbers are more elusive than we think

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• 1k
The numberline shows us an order, and this order gives zero a place. But zero has no place within an order, because it would mean that there is a position of no order within that order, which is contradictory.

:down:

You are getting lost in metaphors and intuitions, as if a checkmate is illegitimate because involved the bishop was never baptized.
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accommodates increasingly general uses of arithmetic, which in my opinion and following Wittgenstein's general philosophy, is best understood in terms of games of increasing generality .sime

:up:
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I still don't see the confusion over negatives and their operations, but then I "do" math every day. Oftentimes familiarity makes it difficult to see how others must view the same.

Consider also the joys of being a crank. If I can make a case that all the geniuses got it wrong, then what's that make me ? "All of math is a contradiction [,and only the great Me can see it]." To be fair (sounds like we've both taught/teach math), students (who want a grade to get a degree to get a job) often actually struggle, thankfully mostly with a humility that makes it possible to correct them.
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rotationPie

So -4 = $i \times i \times 4$. A u-turn of whatever +4 is; repeat the offense and it's back to square one! :lol:

Nice!
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This is the very problem I referred to. The numberline shows us an order, and this order gives zero a place. But zero has no place within an order, because it would mean that there is a position of no order within that order, which is contradictory. Set theory suffers this problem which I discussed extensively with fishfry, who insisted that a set with no order is a coherent concept.

In common usage though, negative numbers are used to represent quantitative values, and here zero has a justified meaning. So it is the equivocation in usage, between "negative numbers" representing quantitative values, and "negative numbers" representing positions in an order, which causes a problem.

As is obvious to you, zero is hard, but not impossible, to grasp. Remember the calendar starts from 1 AD and so, technically, this year is 2021 Anno Domini! Go figure!
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Most of what you wrote went over me head, mate!

Even then, what seems to be an important pattern caught my eye. Numbers were initially 1 dimensional (the number line) i.e. the reals. Then they became 2 dimensional (the complex plane). Do you discern a pattern? It appears that our number system is as of yet still incomplete!
• 38
I read the beginning of chapter 13 of the book linked. It does mention the -1 : 1 ratio argument, which was put forward by Antoine Arnauld and discussed among "many men", including Leibniz whom agreed there was an objection but still used them to calculate. Perhaps I didn't read far enough, but the person mentioned that believed negatives were greater than infinity was John Wallis, who actually did accept them, but thought so because dividing by 0 gives infinity, and going smaller would have to mean going past infinity. Strange indeed.

I don't quite understand the counter to analyzing past mathematicians views on this though. Why else do we study philosophy, especially the ancients? Historical inventory for sure, but we often learn things ourselves by studying their thoughts (of course, sorting the good ideas from the bad). Furthermore, history does repeat itself. It seems we accept negative numbers now on a similar footing as whole numbers, but complex numbers are still pretty hotly debated as to whether we should consider them as real as the real numbers. Whether or not it's an important issue that requires support from the mathematical community, that's why I'm on a random philosophy board and not writing letters to my local university or something.
• 1k
So -4 = i×i×4i×i×4. A u-turn of whatever +4 is;

:up:

Note that you can also go the other direction too using $-i$.
• 1k
It appears that our number system is as of yet still incomplete!

https://en.wikipedia.org/wiki/Octonion
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but thought so because dividing by 0 gives infinity, and going smaller would have to mean going past infinity. Strange indeed.

I think this supports my case that the lack of formalization of real analysis was part of the problem. Of course we can't divide by zero, but we can take limits, which is more reliable when they've actually been strictly defined. Weierstrass is one of my heroes.
• 38
My primary explanation was sort of that too, that it was when we stopped treating mathematics as uncovering truth about the world or as something real, and more as a formal set of rules that we stopped treating negatives as something spooky. It really did unleash a mathematical beast with that change in perspective that allowed us to do math in ways we never would have thought up before. (At least, that's wht I'm assuming based on what I read about the period)
• 1k
we stopped treating mathematics as uncovering truth about the world or as something real, and more as a formal set of rules that we stopped treating negatives as something spooky.

:up:

I dabbled in nonstandard analysis for a week or two, but induction on hyperreal integers was so unintuitive that it felt like cheating. In case you haven't looked into it, a hyperreal is an equivalence class of sequences of real numbers which are equal on an ultrafilter (a weird kind of subset of $\mathbb{N }$, and you can't actually construct one of these ultrafilters but only prove they exist somewhere out there.) Anyway, there are all kinds of infinities and infinitesimals in this system-- and it's just as solid as the ordinary real numbers logically. Cool..but I couldn't take it seriously, lost heart. Too weird, too unreal.
• 2.7k
Perhaps I didn't read far enough, but the person mentioned that believed negatives were greater than infinity was John Wallis, who actually did accept them, but thought so because dividing by 0 gives infinity, and going smaller would have to mean going past infinity. Strange indeed.

You are correct. This is revealed on page 253 from a work by Wallis in 1655. My reference is later in the book, page 593, concerning what Euler thought in 1750+.

It seems we accept negative numbers now on a similar footing as whole numbers, but complex numbers are still pretty hotly debated as to whether we should consider them as real as the real numbers.

Good point. Apokrisis mentioned this in a previous post, regarding Penrose's fascination with complex numbers.
• 10.9k

Most mathematicians seem to just take zero for granted, with zero understanding of what "zero" means. But of course, as I explained, the meaning of "0", as it is commonly used by mathematicians, is ambiguous.
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:up:
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Most mathematicians seem to just take zero for granted, with zero understanding of what "zero" means.

How true. It means nothing to me. :sad:
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Most mathematicians seem to just take zero for granted, with zero understanding of what "zero" means. But of course, as I explained, the meaning of "0", as it is commonly used by mathematicians, is ambiguous

How abso-fucking-lutely fascinating! :up:
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But of course, as I explained, the meaning of "0", as it is commonly used by mathematicians, is ambiguous.

Only in the sense that they have so many exact, formal systems that successfully employ zero that you'd want to know which successful specification of the concept was context relevant.

It's like the cat calling the potty blank when metaphysicians chide mathematicians for ambiguity
• 10.9k
Only in the sense that they have so many exact, formal systems that successfully employ zero that you'd want to know which successful specification of the concept was context relevant.Pie

If your goal is deception, ambiguity is a very useful tool. Therefore successful employment of the term does not indicate that the term is not ambiguous.
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In the Euclidean or complex plane what happens in the vicinity of zero can be far more mind-blowing than MU imagines. In the complex plane the function $f(z)={{e}^{\frac{1}{z}}}$ exhibits a bizarre behavior in that, for any complex value, w, and within any tiny circle centered at zero, there exists an infinite number of values of z such that $f(z)=w$.

Edit: Whoops, gettin' old. There are two points that are exceptions: zero and infinity.

Zero is an essential singularity of this function.
• 10.9k

It is mind-blowing if you're into that stuff, but I'd say it's not at all surprising.
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Cool fact.

I never seriously explored complex analysis, but I'm dimly aware of some spectacular theorems.
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It is mind-blowing if you're into that stuff, but I'd say it's not at all surprising.

What's :scream: to one is just :yawn: to another, eh? I quite like that! The most likely reason is that, sensu amplo, some of us are from another frigging planet/time! Been there, done that! :meh:

It seems important to distinguish difficult from boring, oui mes amies?

:snicker:
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Moderators: How do I eliminate multiple posts?
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↪jgill

It is mind-blowing if you're into that stuff, but I'd say it's not at all surprising

One of the strangest elementary features of the complex plane is the point at infinity. No matter which direction you go, if you keep moving outward, like beyond an expanding set of circles centered at zero, you approach a "point" at infinity. This is "true" since whatever is out there corresponds via projection to the north pole of the Riemann sphere. :cool:
• 10.9k

There is no point at infinity in the complex plane. That point is by definition outside the plane. To allow it in is to break the rules of the structure. There is no north pole in the Reimann sphere. This is a simple result of the incommensurability between the curved line and the straight line. A tangential line can never actually touch the curved line at a point, because the curved line requires multiple points to express its curved nature, in relation to the straight line. So there cannot be a "point" on a curved line, in the same way there can be a "point" on a straight line. Likewise, there is no centre point of a classical two dimensional circle, as indicated by the irrationality of pi. The one dimensional and the two dimensional are fundamentally incompatible.

This is why zero, like infinity, has no place within ordinal numbers, and must be excluded. "Order" is something other than the numbers which represent it, and at those supposed points, zero and infinity, which mark the ends of the order, the represented order is excluded.
• 2.7k
There is no point at infinity in the complex plane. That point is by definition outside the plane. To allow it in is to break the rules of the structure. There is no north pole in the Reimann sphere

This is why zero, like infinity, has no place within ordinal numbers, and must be excluded.

I admire your certitude. It must be nonplussing to watch the world of science evolve using a flawed intellectual mechanism. :confused:
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There's a lot more than one flawed intellectual mechanism out there. And what has happened is that these flawed intellectual mechanisms have led us to dead ends in the evolution of the world of science.
Dead ends are where our attempts to understand can go no further due to the faulty principles being applied, like the dead end which has been reached in quantum mechanics. Dead ends are an integral part of evolution because evolution is a process based in trial and error. The dinosaurs got bigger and bigger, but bigger wasn't necessarily better. In the case of the evolution of scientific understanding, we get the opportunity to look back and find those faulty intellectual mechanisms, and how they led us in the wrong direction.
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