How Do Neutron Stars Relate to the Concept of Negative Numbers in Mathematics?

[Hurkyl]

So as not to hijack this thread, I'm responding here.

First, an obligatory opening potshot.

I too fully understand where the pure mathematicians are coming from.

Isn't that difficult, since you're not a pure mathematician?


Rather than going over the other stuff, I'm going to hone in on this one point:

I fully accept the notion of negative numbers and imaginary numbers. ... Like as if that concept has merit without any relation to anything else.

You are apparently talking about one of the symmetries of an ordered group: if you reverse the order, you're left with the "same thing". IOW, you have to "choose" a direction before using such a thing.


Now, I would agree with some of the points you're making. Some applications of real numbers do require a "choice" of direction, such as a common freshman quandary of whether acceleration due to gravity is a positive or a negative number.


However, you neglect to account for many applications of multiplication. The fact that 1*1=1 forces a unique choice of order, because the product of two negative numbers must be positive.

As a physical example, recall that, in Euclidean space, the dot product of two vectors is an invariant quantity -- it doesn't depend on any choices. The dot product of oppositely pointing vectors will be negative, period.

Another is that of scale factors. There is a "physical" difference between scaling by 1 and by -1 that, again, cannot be explained away by any sort of argument that it's all "relative".

 

[Hurkyl]

On "absoluteness": using my best guess of what you mean, you simply don't know the definitions.


For example, an ordered group is a triple (G, +, <) that satisfies some axioms, such as a < b implies a + c < b + c.

(remember that G, +, and < here are just symbols -- they're not required to be the addition and order relations of the real numbers. For example, (\mathbb{Q}^+, *, &gt;) is an ordered group)

And if (G, +, <) is an ordered group, then so is (G, +, >). (Where > is defined by a < b iff b > a)


So, as you see, when specifying an ordered group, one specifies the order, rather than assuming it's some inherent property of the underlying set.


But, as I mentioned, when you're dealing with a ring, like (Z, +, *), then if it can be ordered, then the multiplication operation fixes the order, so there is a unique relation < such that (Z, +, *, <) is an ordered ring.


In fact, there's a theorem that says for any formally real field, there is a unique relation < that turns it into an ordered field.

(A field is a commutative ring with division, and it's formally real if x_1^2 + x_2^2 + \ldots + x_n^2 = 0 implies all of the x_i are zero. An example of a formally real field is the rational numbers)

I think this theorem doesn't require a field (i.e. doesn't require division), but I'm not sure.

 

[NeutronStar]

To begin with, I openly admit that I'm not a pure mathematician (even though I sometimes feel that way because I've been studying more math than physics lately ).

On "absoluteness": using my best guess of what you mean, you simply don't know the definitions.

It's not that I don't know the definitions. Recall that the other thread was a discussion on whether mathematical concepts are invented or discovered.

I was merely saying that some of the primitive definitions in mathematics are ontologically incorrect (and therefore they are arbitrary human inventions).

Once, a person accepts those primitive definitions I'm sure that they can prove anything that those definitions define.

If the primitive definitions, or axioms, state that empty sets and negative sets can exist as independent mathematical objects, and concepts like fields, rings and groups are all based on those primitive axioms, then those new higher-level concepts are only going to support what the original axioms have already proclaimed.

So I really don't see the point in trying to prove the ontological correctness of the primitives by appealing to higher-level concepts of mathematics like rings or fields that have their foundations based those very same questionable primitives. That's kind of circular isn't it?

Just for the record, you are right that I'm not a pure mathematician. I'm not real familiar with all of the definitions of Abstract Algebra and Group Theory. I wish I were. I'm actually studying those subjects right now. Unfortunately I'm doing this as self-study which makes it that much slower and more difficult. Plus I'm doing this concurrently while studying the Mathematics of Classical and Quantum Mechanics (that's actually the title of the book that I am using as a hub for my studies).

In any case, I don't see how any higher-level topic can possibly retroactively repair the ontological incorrectness of the fundamental axioms. If the primitive axioms of mathematics are ontologically incorrect then all of the higher-level concepts that rely on them are also going to be ontologically incorrect as well. You can't use those higher-level concepts to prove the ontological correctness of the primitive axioms. So to be quite honest with you I don't understand what you're trying to prove by your posts in this thread.

 

[Hurkyl]

Recall that the other thread was a discussion on whether mathematical concepts are invented or discovered.

The reason I split this off into another thread is because I did not intend to discuss that topic.



First off, as far as I can tell, it's irrelevant if the "primitive axioms" are "ontologically incorrect" -- at some level, there's little difference between representing "things" as some combination of mathematical entities, and as some combination of letters.

Secondly, I'm not sure upon what you base your assertions that the "primitive axioms" are "ontologically incorrect" -- the "primitive axioms" of mathematics don't come with any sort of claim that they refer to reality.


Thirdly, this is a red herring anyways -- I was responding to your latest diatribe against the notion of negative numbers. Whether they're founded on set theory, category theory, or even themselves taken as fundamental is irrelevant.


Recall you made some nebulous claims about negative numbers being "relative" or "absolute", and for some reason, you found "relative" to be acceptable, but "absolute" not.

I took my best guess at what you mean by "relative" and "absolute" (actually, I took two guesses).

(Actually, I had three guesses, but I didn't think it worth pointing out that the "absolute value" of a number is never negative. )


Would you care to respond to my criticisms about your latest argument against negative numbers?

 

[Les Sleeth]

First off, as far as I can tell, it's irrelevant if the "primitive axioms" are "ontologically incorrect" -- at some level, there's little difference between representing "things" as some combination of mathematical entities, and as some combination of letters.

I can see you are debating his point differently than the question I want to ask you. You are saying ontological correspondence is irrelevant, but aren't there ontological correlates to negative numbers? What about the effect on my view while looking through a telescope backwards, or that mirror on the left side of my car with the words written on it "objects appear smaller than they are"?

 

[arildno]

Les Sleeth:
I, for one, certainly agree with your view that there exist types of objects/structures which most naturally/easily can be modeled by the introduction of negative numbers (I liked your convex/concave example).

However, I think that the possible modelling capacity of some mathematical objects has little, if any bearing upon the question:
Are they legitimate objects of study logically/mathematically?

I would, however, contend, that those branches of mathematics which most easily may be used in modelling, have a certain charm or interest that those branches further removed from the domain of applicability lack (that's a subjective opinion, of course)

 

[Hurkyl]

You are saying ontological correspondence is irrelevant, but aren't there ontological correlates to negative numbers?

I'm saying that it doesn't really matter if the "primitives" used to "build" the negative numbers have an correspond to anything.

 

[Les Sleeth]

I would, however, contend, that those branches of mathematics which most easily may be used in modelling, have a certain charm or interest that those branches further removed from the domain of applicability lack (that's a subjective opinion, of course)

I know that's true for me, being a concrete and pragmatic sort of thinker. BUt I think I understand how expertise in something might encourage exploration which may not have any immediate application.

 

[NeutronStar]

The reason I split this off into another thread is because I did not intend to discuss that topic.

Well, that's fine, but the comments that I posted in the other thread were related to that topic. So I'm not sure if they would apply to the context in which you'd like to discuss them.

The following may help to clear this up a bit,…

First off, as far as I can tell, it's irrelevant if the "primitive axioms" are "ontologically incorrect" -- at some level, there's little difference between representing "things" as some combination of mathematical entities, and as some combination of letters.

Secondly, I'm not sure upon what you base your assertions that the "primitive axioms" are "ontologically incorrect" -- the "primitive axioms" of mathematics don't come with any sort of claim that they refer to reality.

In the other thread I tried to make it clear that I was coming from the point of view of the following conditional statement:

IF mathematical formalism is supposed to correctly represent the ontological quantitative nature of our universe, THEN our current modern mathematical formalism is ontologically incorrect.

Based on your quote above you seem to believe that the hypothesis of my conditional statement is false (i.e. it's unimportant whether mathematics ontologically correct in its description of the quantitative nature of the universe). If that's the way you feel then why do you concern yourself with the truth value of the conclusion of my conditional statement? It would seem to me that the conclusion of my conditional statement would be irrelevant to anyone who believes that the hypothesis is false.

Would you care to respond to my criticisms about your latest argument against negative numbers?

First of all, let me make it clear that I'm not against the idea of a number having a relative negative property associated with it. (In other words, from a layman's point of view I would still appear to accept the idea of negative numbers as we normally think of them).

The problem I have is with the formal definitions. Mathematical formalism currently allows (or even dictates) that the negative property of a number is actually a part of the number idea itself. I submit to you that this is ontologically incorrect, and therefore it is technically incorrect.

The negative attributes of ontological quantities are always relative. There are no absolute negative quantities in the universe. So in that regard mathematical formalism is ontologically incorrect because it allows for such mathematical objects (i.e. negative numbers) to exist in their own right apart from any relative relationship.

As to your criticisms and examples, I really don't want to get bogged down in the details of each example, but I will say this,...

Most of the examples that you gave in your first post refer to concepts that require a coordinate system. The negativity arises in these cases from the arbitrarily chosen origin of those coordinate systems. Scaling, for example, is only positive or negative relative to whatever scale you are considering to be your starting point. So Scaling is always relative to your starting point. Vectors don't even make sense outside of a coordinate system and that that speaks for itself, unless you want to view vectors from the point of view of pure Abstract Algebra (see next paragraph)

In your second post your refer to concepts such as fields, rings and groups etc. These are the concepts of Abstract Algebra and Group Theory which are themselves based on the primitive axioms that have already proclaimed that negative numbers can exist as independent objects. Therefore, these higher-level abstract formalisms are of course going to support, and further develop, those basic assumptions.

All I'm saying is that logically consistent primitive axioms can be constructed in a way that treats negativity as being a separate concept from the quantitative concepts called numbers. Had this been done properly with the original primitive axioms, then modern Abstract Algebra and Group Theory would indeed be built on those axioms and support those ideas.

So I'm just saying that it is possible to go either of these two routes and still have a consistent logical formalism. The only difference is that one of these formalisms will me more ontologically correct than the other. So why aren't use using the more ontologically correct formalism?

The answer seems to be that somewhere along the way the mathematical community decided that being ontologically correct is not important. That is when they made what I consider to be a "wrong turn".

This actually took place historically in about 1850 with the beginnings of the formalization of Set Theory, and the introduction of the formal definitions of the idea of number.

Prior to 1850 the idea of number was pretty much taken for granted as merely being entirely intuitive. It was during the formalization of the definitions of numbers and in particular the definition of the [i[Natural Numbers[/i] where the mathematical decided to move away from ontology and toward a purely abstract logical system based solely on arbitrary human inventions. In fact, the bulk of these arbitrary ideas were invented by a mathematician named Georg Cantor.

I'm actually writing a book on all of this and I don't intend to re-type the entire book in here. Especially to an unreceptive or hostile audience.

In short, if you don't believe in the hypothesis of my conditional statement given earilier in this post, then you won't be interested in reading my book. I actually give that conditional statement early in the book and explain to the readers that if they don't believe that mathematics should be ontologically correct then there is really no sense in reading any further.

In my book I present the idea of negativity as being akin to an adjective in the language of mathematics while the idea of number is more akin to a noun. That's probably the best way to put it for clairty in laymen's terms.

Current mathematical formalism has negativity defined as a noun, when in reality it is actually an adjective. The mathematical operations are more akin to verbs. In my book I break mathematics down into the formal logical language that it is. I do that for the laymen's sake, and for clarity. But I also emphasis the importance of logical consistency, so there is also some symbolic logic in the book as well, not to mention the normal symbols of mathematics and their definitions and meaning too.

In the meantime I'm really not interested in arguing my case on Internet forums. Especially to people who are hostile toward them. I post some of my ideas from time to time in related threads to see if there is any positive response from like thinkers. Unfortunately I don't get much positive response. Most people are bent on defending the current state of mathematical formalism at all costs. Almost to the point where they are completely unwilling to even begin to imagine that there just might be a better way to construct it.

So to answer your question in breif,... No, I have no interest in trying to convince you of something that you are not interested in being convniced of.

Let's see,... Now I'll only sell 999,999 copies of the book instead of a million because Hurkyl isn't buying into this.

 

[Hurkyl]

In the meantime I'm really not interested in arguing my case on Internet forums.

Ah, so you're preaching. Even better.


How much do you know about the proof that you can't prove the parallel postulate of Euclidean geometry? AFAIK, that was the first example of the utility of modelling.


I'm not sure what you mean by "negativity is a noun": "x is negative" is a unary relation, the most direct logical analog of an adjective.

 

[NeutronStar]

Ah, so you're preaching. Even better.

Exactly when did the presentation of a logical argument become known as preaching?

I start out with a conditional statement. I give the reader a chance to bow out gracefully if they believe that the hypothesis of my logical statement is false. However, if they believe, as I do, that it at least should be true, then I present what I believe to be a very convincing logical argument why the conclusion of my conditional statement must then also be true.

So that's what preachers do nowadays? I wasn't aware of that.

Actually I might point out that you started this thread with my screen name in the title. I didn't start any thread to preach my discoveries. I did post my thoughts in a thread that questioned the differences between discovering things and inventing them. Georg Cantor invented the empty set. I discovered that Cantor's ideas are both logically and ontology incorrect.

Anyhow, calling me a preacher is hitting below the belt!

It's also totally unwarranted.

How much do you know about the proof that you can't prove the parallel postulate of Euclidean geometry? AFAIK, that was the first example of the utility of modelling.

This is precisely why I'm writing a book about it rather than trying to discuss it publicly. What do parallel lines have to do with whether or not negativity is modeled correctly? Parallel lines have to do with geometry and ideas of infinity. Neither of those higher-level ideas need to come into play here so why are you even bringing them up? Sounds like a complete distraction tactic just so you can make some kind of point that is totally irrelevant to anything that I was saying, and then make it look like as if you proved some kind of point.

I'm seriously not interested in arguing with someone who is so hostile to my ideas. If you believe they are hogwash so be it. No need to call me a preacher.

Why don't you just accept that you disagree with my hypothesis (that mathematics should be ontologically correct) and leave it at that?

Are you going to try to convince me that mathematics should not be ontologically correct, or that any attempt to try to make it ontologically correct is futile because ontological formalisms are an illusive dream that cannot possibly be attained?

I believe otherwise.

 

[Hurkyl]

Preaching is what some call it when you make assertions, but have no interest in arguing for them.

What do parallel lines have to do with whether or not negativity is modeled correctly?

Nothing. But as I recall, this proof was historically the first example of the concept of a model. And, quite appropriately, it appeared right around the period of time you claim mathematics went astray.

The history of noneuclidean geometry gives an important perspective on the developments of this time. The philosophical consequences of Beltrami's proof were very significant. No survey of the developments of the time period could be complete without accounting for this work.

Anyways, this particular topic was brought up because I hope it would be helpful to you.


I forgot to respond to this last time:

Most of the examples that you gave in your first post refer to concepts that require a coordinate system. The negativity arises in these cases from the arbitrarily chosen origin of those coordinate systems. Scaling, for example, is only positive or negative relative to whatever scale you are considering to be your starting point. So Scaling is always relative to your starting point. Vectors don't even make sense outside of a coordinate system and that that speaks for itself, unless you want to view vectors from the point of view of pure Abstract Algebra (see next paragraph)

Wrong. I even specifically chose these examples because they are coordinate-free.

Les sleeth gave particular examples of scaling factors, taken straight from optics. The lateral magnification of a lens or mirror does not depend, in any way, on a choice of coordinates.

My other example, of a scalar invariant, I chose specifically because of its importance to physical applications. Recall that it is useful precisely because it does not depend on a choice of coordinates.