Handling mathematical concepts that have no relation to reality

Science & Technology Physics and Mathematics
101

OK, Icosahedron, I’ve been reading your posts for several days now and I believe I am finally beginning to understand your system. I still disagree with it, mind you, but I think I might understand it. I’ve got some questions that migt “gel” that understanding.

Looking at integers. If (1,0) represents what I think of as +1, and (0,1) represents what I think of as -1:

a ) what does (0,0) represent?

b ) what does (1,1) represent?

c ) Is it legal to ever put a negative number in one of the indices, e.g, (-1, 0)?

d ) Is it a requirement that at least one of the two numbers in the pair be zero?

e ) What would (3,2) represent (assuming it’s legal)?

a) (N,N) represents 0 for any natural number N

B) See response to point (a)

c) No. One of my tenets is we only need natural numbers, and ratios of them, in order to record measurements.

d) Yes and no. Depends on what you are doing. For example, when subtracting 4 from 3, the steps would be:

(3,0) - (4,0) = (4,1) - (4,0) = (0,1)

The non-reduced representation, (4,1), is useful in doing computations without encountering numbers less than zero. This is analogous to how addition of 1/3 and 5/6 goes like:

1/3 + 5/6 = 2/6 + 5/6 = 7/6

e) (3,2) is a non-reduced version of (1,0)

  • David
102

Nate T said: “Ah, I see, icosahedron. Your argument is that we have nothing but (positive) rationals to measure with in reality (which is true), and therefore we should have no need of real (or negative) numbers in mathematics?”

In a nutshell, yes. Given that every other facet of our society is less than wholly rational, why would one expect mathematics as it stands to be pristine?

Now, if one chooses to identify limits of non-terminating sequences of rationals with special symbols, fine. Not sure what the point is of identifying them for metrical purposes, however.

On the other hand, if one discovers solutions to equations in algebra, and then insists on conflating them with rational measurements from reality, to form the so-called real algebraic numbers, then one will have to drop some context to get the scheme to work in practice. And that’s not good, long range – it makes the math muddy at best, harder to learn and teach, and harder to use, forcing humans to “fly by instruments” instead of using their conceptual peepers.

The context dropped is whether or not the idea measurably exists in reality. As usual in any invalid compromise, it is the reality basis that gets cut off in such compromise.

Perhaps one wants to take the ratio of a circle’s circumference to its diameter, in the limit of an unreal, perfect circle (measuring any real “circle” will yield a rational approximation to PI, of course). In this case, even beyond the irrationality of PI, I question the premise that a curved segment is commensurable with a straight segment, except for the (rational) measurement of how much stuff they are made of.

In this case, the context dropped is that angles have units (they are fractions of a cycle), and one should properly say that a one meter radius circle has a circumference of one meter-cycle.

Cheers.

  • David
103

Aha! This objection I understand. You should have picked the moniker Pythagoras, or maybe Kronecker. :stuck_out_tongue:

Now, if one chooses to identify limits of non-terminating sequences of rationals with special symbols, fine. Not sure what the point is of identifying them for metrical purposes, however.

That’s because people are generally taught all of these extensions of numbers (like negative, real, and complex numbers) as a system of rules apart from the context from which they arose. If they actually taught people what real numbers *are*, things would be different.

On the other hand, if one discovers solutions to equations in algebra, and then insists on conflating them with rational measurements from reality, to form the so-called real algebraic numbers, then one will have to drop some context to get the scheme to work in practice. And that’s not good, long range – it makes the math muddy at best, harder to learn and teach, and harder to use, forcing humans to “fly by instruments” instead of using their conceptual peepers.

If you handle only rational numbers, how are you supposed to solve x^2 = 2? Do you content yourself with saying that you can find a sequence of rationals that solves the equation as precisely as you’d like? Because when you unpack the definition of the real number \sqrt(2), that’s what it amounts to.

The context dropped is whether or not the idea measurably exists in reality. As usual in any invalid compromise, it is the reality basis that gets cut off in such compromise.

I agree that “reifying” real or negative numbers (such as claiming a stick has length exactly \sqrt(2), or that there be -1 cows in a field) is silly.

But in fact I don’t think people generally make the kind of gross category errors you subscribe to them, and I think it’s possible to keep in mind what these concepts represent in whatever context you’re applying them to.

For example, reifying real numbers is generally harmless in applications since people pass to approximations at the end anyway, and I’ve never met a student so thoroughly confused by negative numbers to seriously think that the use of “-1” implies the existence of negative numbers of objects in reality.

104

One use of negative numbers that doesn’t get mentioned is when measuring the “opposite” way from a referent as is usually done. For example, altitude below sea level, or degrees below zero (temperature). These are somewhat tangible uses. And even with regard to “owing money” a negative net worth (owing more than you own) is meaningful in fact.

105

If folk would stop with the demand for compactness,

I’m trying to simplify things, believe it or not.

You fail at simplifying. The need for compactness, also referred to in Objectivism as the principle of unit-economy, is real not arbitrary.

Subtracting 4 from 3 with the steps: (3,0) - (4,0) = (4,1) - (4,0) = (0,1) using an intermediate “non-reduced representation (4,1)” is not a gain in simplicity. You want to trade off the complexities of one notation system for another, and it is not a good trade.

106

One use of negative numbers that doesn’t get mentioned is when measuring the “opposite” way from a referent as is usually done. For example, altitude below sea level, or degrees below zero (temperature). These are somewhat tangible uses. And even with regard to “owing money” a negative net worth (owing more than you own) is meaningful in fact.

Meaningful as vectors, yes. Why not? That’s what they are (words like “below” indicate a direction).

107

You fail at simplifying. The need for compactness, also referred to in Objectivism as the principle of unit-economy, is real not arbitrary.

Subtracting 4 from 3 with the steps: (3,0) - (4,0) = (4,1) - (4,0) = (0,1) using an intermediate “non-reduced representation (4,1)” is not a gain in simplicity. You want to trade off the complexities of one notation system for another, and it is not a good trade.

As Nate T. has pointed out, I mean continuous, not compact. Sorry for any confusion.

It isn’t a gain in simplicity on its own. But it makes it easier to take the next conceptual steps towards one of my goals, a wholly rational mathematics where unreal ideas such as “less than nothing” and “continuously variable” are left by the wayside.

  • ico
108

If you handle only rational numbers, how are you supposed to solve x^2 = 2?

I consider the space of solutions to algebraic equations to be perfectly fine, don’t question it per se.

My problem is, why does it need to be conflated with a one’s metric space, just because the two correspond at some points?

Why must the second root of 2 (obtained originally as a solution to an algebraic equation) be considered on the same basis as PI, which is not algebraic?

I think the conflation is an historical artifact from the days before fast computers, which can’t handle floating abstractions.

  • ico
109

Nate T., or anyone else who wants to handle it:

If, as I claim, magnitude and direction are the definitional attributes of the concept “vector”, then:

  1. Integers are vectors

  2. What sense does it make to multiply vectors?

  3. How does (-1)*(-1) = (+1) make any sense? Said in English: how does adding -1 copies of -1 to yield +1 make any sense?. Huh?

  • ico
110

Re #3, when you multiply a number by 1, you get the same number. When you multiply it by the opposite of 1, you should get the opposite of the number you started with. So when multiplying any positive number by -1, you get a negative number, its opposite, e.g. (-1)*27 = -27, its opposite. Similarly, multiplying a negative number by -1 should return the opposite of the negative number. Since the negative number is itself the opposite of the positive number, it stands to reason that the positive number is the opposite of the negative number, so e.g., (-1)*(-27) = 27. And of course (-1)*(-1) = 1.

“multiplying vectors” there are two ways to do this of course, the cross product and the dot product. If you regard integers as vectors of order 1, (which is a more common way of doing this than your system), you can take the dot product of them and it works: (-2) dot (3) = -6 which is really (-6). That actually DOES make sense, but only because I did it the standard way. By insisting on breaking integers into vectors of order 2, you successfully make it not make sense; the same problem is now rendered (0,2) dot (3,0) = 0+0 = 0, or rather (0,0). So dot-multiplying numbers of opposite signs under your system gives you a result of zero, without fail. This is a silly result, IMHO. But a dot product breaks down in another crazy fashion when you are mutiplying two negative numbers together. (0,2)dot(0.2) gives 4… but somehow, that’s really supposed to be (0,4) or -4 to us non-icosical math types. Actually the dot product accidentally gives the right answer which is 4, but you reject it out of hand because you don’t think it makes sense. Unless of course multiplying vectors isn’t done with a dot product in your system. (But if not, how is it done? Or are you claiming you cannot multiply vectors at all? If not, since all numbers are vectors according to you, how can you multiply them at all?)

Now I can reconcile “all integers (or rationals, or reals) are vectors” IF they are vectors of order one. As soon as I try to do things your way, I get nonsensical results, some of which you actually complain about being nonsensical–multiplication not making sense for instance. That nonsensicality is a result of fundamental problems with your premises; they are a logical consequence of your error in trying to treat negative numbers as orthogonal to positive numbers, when they in fact aren’t independent of each other. Never mind the fact that (1001,1000) and (1,0) are the same number supposedly, yet have totally different magnitudes! Hmmm, in the normal system, there’s one way to represent 3, and that’s “3”. (or you give someone an arithmetic problem to solve.) Under your system there are ONLY arithmetic problem methods of representing 3. So perhaps it is you who should check your premises. For one thing a number line is not two rays, it is number line with some point on it designated as zero and the points to one side considered to be opposite in sign (but in the same dimension) as the points on the other side. (Else why must the two rays you think it consist of be at a 180 degree angle from each other?)

And OBTW contrary to your assertion several posts back, two rays sharing the same starting point with an angle between them other than 0 or 180 are all you need to define a plane, it doesn’t take three. A (superfluous) third ray would have to be coplanar with the first two (much as your superfluous second ray for a number line must be colinear with the first one). So much for your system being simpler than the conventional one. Of course it gets awkward specifying the positions of things on the plane with those two rays but of course if you’d just use negative numbers (and extend the rays to lines) that problem would disappear.

In fact the more I think about it the more insane your assertions of your system being simpler become. The conventional system uses one number to specify an integer. Yours requires two. And if one of them isn’t zero, you can and ought to “reduce” it. The fact that (0,3), (1,4) and (2,5) all represent the same thing tells anyone with a lick of sense that your system has needless redundancy in it. The conventional system uses a pair of numbers to specify a point on a plane. Yours requires three to specify positions on three rays, which had better be coplanar… and I haven’t asked but I bet those triplets can and ought to be reduced so that at least one number is zero–again pointing to redundancy.

I believe this should be enough to convince anyone except you (since you have a fundamental problem with negative numbers and have erected this elaborate scheme to try to get around them–then complain when the results make no sense) that your system is fundamentally bogus. As I said, nonsense on stilts.

111

Re #3, when you multiply a number by 1, you get the same number. When you multiply it by the opposite of 1, you should get the opposite of the number you started with.

What exactly is “the opposite of 1” anyhow? Opposite in what sense? In the sense that by “1” you mean “+1”, i.e., opposite as vectors? In which case you haven’t defined multiplication of vectors yet …

“multiplying vectors” there are two ways to do this of course, the cross product and the dot product.

The dot product results in a scalar, not a vector, so is not a proper group-theoretic multiplication operation.

The cross product is not commutative, so it doesn’t meet the requirements.

If you regard integers as vectors of order 1,

No, the naturals are vectors of order 1 in the sense you mean; to have a vector, you need an origin. And if the origin is the integer 0, then -1 and +1 lie on different rays from the origin, i.e., different vectors.

So dot-multiplying numbers of opposite signs under your system gives you a result of zero, without fail. This is a silly result, IMHO. But a dot product breaks down in another crazy fashion when you are mutiplying two negative numbers together. (0,2)dot(0.2) gives 4… but somehow, that’s really supposed to be (0,4) or -4 to us non-icosical math types.

Why are you surprised that my inner product works as intended? I have already made clear that I don’t grant the possibility of moving left by moving right – or, to put it more concretely, you can’t get to the moon by heading towards the center of the Earth. Left/right, in/out are orthogonal conceptually and existentially in the simplest sense: displacements in one direction cannot be added up to form displacements in the other.

Unless of course multiplying vectors isn’t done with a dot product in your system. (But if not, how is it done? Or are you claiming you cannot multiply vectors at all? If not, since all numbers are vectors according to you, how can you multiply them at all?)

I never claimed all numbers are vectors. I simply claimed that vectors are not numbers, and that integers are a form of vector by definition: magnitude and direction. I don’t know how to define multiplication on vectors in a group-theoretic sense …

Not sure why you can’t bring yourself to use +3 when you mean the integer, and 3 only when you mean the natural number. If you’d do that, you’d be representing the integers as the cross product of the naturals with the cyclic group of order two – which is also a vector form, albeit not symmetric in its coordinates (which is where I started, btw – but quickly realized the benefit of the symmetry in terms of simplicity and ease of use). In other words, instead of -3 and +3, you could write (-,3) and (+,3) – what’s the conceptual difference. Why do you insist on conflating the positive integers with the natural numbers? They are clearly different concepts, arrived at by distinct means, so why obfuscate their difference?

Its magnitude AND sign that are needed to specify an integer. Sign has two states, which is no coincidence because the integers encompass two rays in distinct directions. And a number line certainly IS two rays point oppositely from the origin, why do you want to ignore the fact that the origin is special. Ah, maybe that’s the problem – you want the luxury of moving your origin along the line without loss of generality. But the origin is just a point of perspective, and with respect to any such point, the line is seen as a ray when one looks left, and a ray when one looks right – two rays, yes.

Oh, and the rays need not be at 180 degrees for my math to work, in fact the math works (and you wouldn’t question it for any sequence of angles less than 180 … but if 180 is somehow special, then you are claiming that the process of widening an angle is not uniform, so that the limit of the process can have properties unpredictable from the properties at any given angle less than 180. Where do you draw the line, so to speak?

It takes two rays to define a plane, but three to span it, which is what I said. If I used “define” where I meant “span”, mea culpa.

Finally, Steve, regarding your repeated swerves towards dubbing my assertions “insane” without understanding my analysis in full, how about a little Shakespeare?: “m’lady doth protest too much, methinks.”

The conventional system uses one number plus one sign to specify an integer.

This is one of my issues: when you conflate 1(natural) with +1(integer), and refer to them with the symbol “1”, and when the magnitude of +1 is equal to 1, its too easy to lose sight of what is really going on.

And, Steve, you have lost sight of it. Check your premises, and please refrain from attacking my character obliquely in the future. I don’t appreciate it, nor do I consider it harmless. I have been respectful to you, but am beginning to accumulate evidence to support a suspicion that you consider my challenge of your premises to be a form of disrespect. If that is the case, let me know and I’ll stop dealing with you altogether.

Cheers.

  • David