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Handling mathematical concepts that have no relation to reality

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brian0918
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Thinking of the solution of x*x=-1 as the square root of -1 is just people mistaking formalism for fact. It is just a number defined by the fact that it solves the equation, and should have a disparate name such as "freddie" to make sure folk don't conflate it with observable, measurable quantity.

Orthogonality is the fact captured by this formalism.

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Mr. D'Ipolito, I respectfully ask that you check your premises.

Any two distinct rays emanating from the same point are linearly independent in the simplest sense: linear displacements along one ray can never get you to a point on the other.

The premise I question is that opposing rays somehow coalesce into linear dependence, whilst rays at a tiny angular displacement from opposition are linearly independent.

The contradiction is due to context-dropping, and a bit of conflation, around the concept "linear". Linear means something different than Euclidean rectitude in the more general context of so-called "linear" algebra, and unfortunately leads to the common mistake of assuming that lines are the basis of dimension in volumetric space.

Volumetric space is the conceptual context. What can be said with certainty about volumetric space? Well, it's finite, bounded, enclosed. In other words, it has an inside, and an outside (editorial note: people traditionally have underestimated the importance of the inside/outside complementation that necessarily adheres to volumetric spaces). Now, the thing about inside and outside is that, like any truly complementary pair, they are not merely symmetric opposites. This becomes clear from a simple observation: you can only go inward so far before you pass through a volume and thenceforth move outward indefinitely (assuming a ray-like trajectory).

In volumetric space, in means in towards something, and directions are defined by reference to "inward-nesses".

Mathematicians may play with more or less unrelated abstractions, but we humans live in volumetric space(s), and cannot assume things are where they are without seeing a signal indicating such facts ... and the signal must traverse volumetric space to reach us ... and two signals sent in different directions are clearly not gonna end up in the same place, clearly represent two distinct dimensions of information gathering. This is just another way of saying, as I originally stated: Any two distinct rays emanating from the same point are linearly independent in the simplest sense: linear displacements along one ray can never get you to a point on the other.

Rays, not lines, are the proper basis of dimension when traversing motion is taken as the gold standard of proof of (potentially evolving) relative location. I challenge anyone to impeach this standard.

Ergo, lines are 2-dimensional.

And therefore, since a vectorial representation is required, I submit my representation of the integers as the most conceptually economic I could devise: pairs of counting numbers with equality defined by reference to equivalent net displacement when one imagines finite, constant-length steps taken to the left, or the right.

I can keep trying to make it clear if you choose to pursue it -- it is worth the effort, I promise.

Cheers.

- David

Edited by icosahedron
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Mr. D'Ipolito, I respectfully ask that you check your premises.

Any two distinct rays emanating from the same point are linearly independent in the simplest sense: linear displacements along one ray can never get you to a point on the other.

Two distinct rays that share the same origin are not actually distinct.

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On reading my prior post, I think it appropriate to give a simple visualization for my model.

Consider a discrete random walk where each step is of the same length and can be taken either left or right.

Let (x,y) represent all paths that subtend x steps left and y steps right.

Notice that (x,y) and (x+q,y+q) refer to the same endpoint in a walk starting at (0,0)

(The number of steps is ignored for this purpose ... but should not be in general, e.g., when planning lunch dates. But that's another, more involved, story of mine; I'll fire it up on a new thread at some point.)

So, I simply take (x,y) to represent the integer y-x, and accept the fact that there are multiple ways to refer to this integer, kinda like the case with rationals where both numerator and denominator can be scaled without changing the value. (Actually, I end up exploiting these multiple ways to model translational momentum, but that is part of the other story).

Cheers.

- David

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Numbers are not rays and should not be modeled as such. By doing so you assume what you are trying to assert.

I do not model numbers as rays; I model quantitative spectra for specifically measurable properties as rays. For example, I model the concept of mass as a ray, with zero as the minimum. Clearly, negative mass is nonsensical. I model the concept "scalar, graded value potential of some measurable quantity" as a ray. Why not? How is that model lacking in representing mass, energy, length, time, or any other measurable quantity? Measurements are always of quantities, which are something, not less than nothing. Measurement often relates to operationally continuously variable quantities, such as a visually observable mass of gold (not always, cf. atomic spectra).

The rays are arrays of numbers ... rational numbers in ascending sequence.

Integers account the distance left or right of zero. -1 means distance 1 to the left of zero. Simple and clear. But this is a vectorial concept -- magnitude and direction is the hallmark of the abstraction mathematician's call "vector". To me, that is also simple and clear.

If integers are indeed vectors, as I claim, then multiplying them is not well-defined.

But adding them is. All I am really doing here is demoting the integers from a ring to a commutative group.

And yes, I understand the repercussions in the organization of mathematical knowledge. But please note, this is a matter of representational truth and efficiency; I'm not saying computations with integers are wrong in appropriate context; I am saying that the concept as currently integrated into the institutional knowledge base is awkward at best, and anything awkward can be improved (otherwise we'd use the word "perfect"). Awkward systems of mensuration stunt the growth of conceptual knowledge by increasing processing time. Put positively, the computer has rendered tables of logarithms obsolete; how could humans have attained the current level of economic capacity using tables of logarithms? An efficient system of mensuration (and hence also representation of computational tasks) redounds forward in time on the cumulative knowledge of humans.

That's why it matters even though the results for already considered cases will be identical. I am after the cases currently occluded by the awkwardness built in to the institutional mensuration conceptioning.

For more fun, note my progression of spatial dimensionality:

1-D: ray

2-D: line (opposing rays)

3-D: plane (rays arrayed triangularly)

4-D: volume (rays arrayed tetrahedrally)

The linear algebra is clear and simple if you consider it a bit.

Cheers.

- David

Edited by icosahedron
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For more fun, note my progression of spatial dimensionality:

1-D: ray

2-D: line (opposing rays)

3-D: plane (rays arrayed triangularly)

4-D: volume (rays arrayed tetrahedrally)

The linear algebra is clear and simple if you consider it a bit.

Um... since a ray is bounded (has a starting point) and a line has another dimension by virtue of its not being bounded, following your "logic", by extension *half* a plane ought to have three dimensions and a whole plane should have four, and a half volume should have five, and a full volume should have six.

And your linear algebra SUCKS if it considers negative numbers to be orthogonal to positive numbers. (Do you know what "orthogonal" means? And what it represents visually?) And do you realize your crazy multiplication example from three posts ago doesn't make sense (as you said) because your premise that a negative number is orthogonal to a positive number is just flat-out wrong, and that it would make sense if you'd pitch that piece of idiocy overboard? If you are in fact justified, then why not consider things in the right half of the plane to be orthogonal to things in the left half of the plane? What justification is there for assuming a line is two dimensional?

I've had my fill of this; it's nonsense on stilts.

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Um... since a ray is bounded (has a starting point) and a line has another dimension by virtue of its not being bounded, following your "logic", by extension *half* a plane ought to have three dimensions and a whole plane should have four, and a half volume should have five, and a full volume should have six.

And your linear algebra SUCKS if it considers negative numbers to be orthogonal to positive numbers. (Do you know what "orthogonal" means? And what it represents visually?) And do you realize your crazy multiplication example from three posts ago doesn't make sense (as you said) because your premise that a negative number is orthogonal to a positive number is just flat-out wrong, and that it would make sense if you'd pitch that piece of idiocy overboard? If you are in fact justified, then why not consider things in the right half of the plane to be orthogonal to things in the left half of the plane? What justification is there for assuming a line is two dimensional?

I've had my fill of this; it's nonsense on stilts.

It is your choice not to deal with me, or my ideas.

Just in case you might still want to deal, I'll show you the formalism, because it is pretty simple.

I find linear algebra, and especially the notion of inner product spaces, to be conceptually unimpeachable as a framework for analyzing problems that can be mapped into a vectorial framework.

Orthogonal in linear algebra simply means that the projection of two vectors vanishes under the chosen inner product. The inner product is a matter of choice, but must be a proper inner product, i.e., it must satisfy the definitional axioms of an inner product.

Formally, then, you ought to have no trouble granting the following vector space construct:

Let P be the set of pairs of natural numbers.

Let R be the equivalence relation defined, for (s,t) and (u,v) elements of P, as:

(s,t)R(u,v) iff s+v==t+u

Let X be the set of equivalence classes of P under the relation R.

Define addition of elements x, y of X by reference to addition of their equivalence class representatives.

The additive identity element is the equivalence class of (0,0).

The equivalence class of (t,s) is the additive inverse of the class of (s,t), because (s+t,t+s)R(0,0) by the commutative nature of addition of natural numbers.

Under this addition, X is a proper vector space.

Note that the simplest, most reduced representative of each equivalence class has one component equal to 0. Thus, (N,0) is the prime representative of the class (N+K,K), and (0,N) for (N,N+K). In fact, all equivalence class prime representatives are of one of these two reduced forms. This is analogous to rational number formalism where 2/3==4/6 where 2/3 is the reduced, prime representation, but 4/6 is equivalent. (In fact, a similar process of considering pairs and forming equivalence classes is applicable when constructing the rational numbers -- but the addition rule is defined differently)

X is seen to be isomorphic to the integers when the following correspondences are noted:

+1 → (1,0)

-1 → (0,1)

Thus it is a perfectly legitimate representation of the integers under addition; it highlights the vectorial nature of the concept “integer”, which is obfuscated in the current +x/-x formalism. The representations are isomorphic, so if one is clearly vectorial, the other must be, too.

So stop trying to multiply integers directly, you can’t do that with vectors!

What you can do is scale vectors and add them. The problem is that the current formalism encourages conflation of the concept “2” as a scalar multiple, with the concept “+2” as a vector displacement.

So, when one speaks like “-2 equals 2 times -1”, what one is really saying is “the vector (0,2) is equal to the vector (0,1) scaled by the scalar 2, i.e., (0,2) = 2*(0,1). My formalism makes it obvious what is going on here, it is a standard linear combination formalism.

Now one can proceed to choose and define an inner product on X, e.g., for the purpose of considering the projection of one vector onto the direction of another. What inner product to use depends on the context one is analyzing.

In the context of modeling spatial relationships, the key conceptual invariant is that spatial displacements are travelable, at least in principle. Since you can’t travel left by going right, I desire an inner product which respects the identification of displacements as travelable, including the fact that left and right are distinctly accessible directions.

The inner product I choose is simply the natural dot product between prime class representatives, so that (1,0) and (0,1) are orthogonal (in the sense of linear algebra, not Euclidean geometry). This fits with observations of physical reality, wherein you can’t ever go left by going right.

It seems strange only because you are used to thinking of lines as the basis of dimensionality.

To represent a plane takes three rays towards the vertexes of a triangle. Cartesian coordinates appear to use only two lines, but they use four rays – three is more efficient, especially so since only positives are necessary to represent the plane in my framework.

To represent a volume takes four rays towards the vertexes of a tetrahedron, not the redundant 6 rays found in Euclidean representations.

It’s a matter of conceptual efficiency, as I said.

If my sense of efficiency offends your sense of orthogonality, then one of us is wrong.

Cheers.

- David

Edited by icosahedron
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Wow. This certainly is a thread about how to handle mathematical concepts that have no relation to reality, all right.

icosahedron, just for the hell of it, since you're representing integers as ordered pairs of natural numbers, just what is this inner product you're defining on the integers? The one that makes 1 and -1 orthogonal?

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Wow. This certainly is a thread about how to handle mathematical concepts that have no relation to reality, all right.

icosahedron, just for the hell of it, since you're representing integers as ordered pairs of natural numbers, just what is this inner product you're defining on the integers? The one that makes 1 and -1 orthogonal?

I agree that considering integers as scalars breaks the tie to reality, that is my point. Integers have magnitude and direction, are vectors.

I map +1 to (1,0), -1 to (0,1), and then use the standard Euclidean dot product of 2-tuples.

I represent +N as N*(1,0), or (N,0); I represent -N as N*(0,1), or (0,N).

When adding, I reduce the result to simplest form using the equivalence relation (s,t)==(u,v) iff s+v=t+u.

For example, I would write the equation (-7)+(+5)=(-2) as (0,7)+(5,0)=(5,7)==(0,2), with the latter step invoking the equivalence relation.

Look, its a perfectly good mathematical invention. It may seem strange to describe a line this way, but it's consistent with reality and linear algebra, and when you go up to planes and volumes, the benefit only increases, conceptually speaking, in terms of simplicity of representation and computation (no negatives to worry about).

I don't expect anyone to take this on faith, but can anybody else see that integers are vectors (have magnitude and direction)? That is all I ask that you grant; the rest of my framework follows logically from that.

Cheers.

- David

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Look, its a perfectly good mathematical invention. It may seem strange to describe a line this way, but it's consistent with reality and linear algebra, and when you go up to planes and volumes, the benefit only increases, conceptually speaking, in terms of simplicity of representation and computation (no negatives to worry about).

I don't expect anyone to take this on faith, but can anybody else see that integers are vectors (have magnitude and direction)? That is all I ask that you grant; the rest of my framework follows logically from that.

Cheers.

- David

1. The fact that algebraic sign encodes the very notion of direction that you're trying to capture with these ordered pairs, and does so in an algebraically unified way, seems quite conceptually and computationally beneficial to me. You just have to keep in mind what negative numbers mean in whatever context you're in.

2. The reason for my opening barb is that you can't use the standard Euclidean inner product on these pairs, since they are really equivalence classes of pairs, as you yourself use. So while (0, 1).(1, 0) = 0, you could also say of the same two integers that (1, 2).(2, 1) = 4. So your operation is not well-defined since you don't get a unique answer for the inner product of two integers.

3. What you are talking about might be better described using positive linear combination. In that case, -1 and 1 are positively linearly independent, as you claim. You might consider checking the notion out on wikipedia (c.f., here).

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1. The fact that algebraic sign encodes the very notion of direction that you're trying to capture with these ordered pairs, and does so in an algebraically unified way, seems quite conceptually and computationally beneficial to me. You just have to keep in mind what negative numbers mean in whatever context you're in.

2. The reason for my opening barb is that you can't use the standard Euclidean inner product on these pairs, since they are really equivalence classes of pairs, as you yourself use. So while (0, 1).(1, 0) = 0, you could also say of the same two integers that (1, 2).(2, 1) = 4. So your operation is not well-defined since you don't get a unique answer for the inner product of two integers.

3. What you are talking about might be better described using positive linear combination. In that case, -1 and 1 are positively linearly independent, as you claim. You might consider checking the notion out on wikipedia (c.f., here).

A1. No disagreement that the traditional formalism is efficient for computations with pen on paper. In the age of computers it hardly matters. And the fact that I have to keep in mind more than necessary, including all the strange rules of multiplication (-1 * -1 = 1 ... wtf?), makes the paper gain not worth the conceptual headache, at least in my mind. As for the "algebraically unified way" you allude to, my system is no less integrated -- it is isomorphic. Finally on this point, if "unified" refers to the rays' endpoints fused to form a line, whether intentionally or not, then it is exactly that notion I am questioning. In reality, I observe visually by reference to my eyes as sensational centerpoint, and all distances are outward and positive relative to my eyes.

A2. The inner product is defined as the Euclidean product of the fully reduced representatives with no loss of generality, i.e., it is a proper inner product on the elements of the vector space of equivalence classes.

A3. Why should I stand aside while a less convenient model continues to co-opt the conceptual space? Linearly independent works just as well for colors as it does for spatial relations, with, e.g., green, red, and blue being orthogonal under an RGB inner product. Since I don't recognize any cognitive primacy of negative numbers as such (beyond some symbolic sleight of hand hiding the fact that they are vectors), neither do I feel it necessary to qualify what I have already, and clearly, shown to be a perfectly good vector space, together with a perfectly good inner product (which I am tempted to dub "displacement product"), and hence perfectly natural sense of linear independence -- plain old-fashioned linear independence, not needing any qualifier such as "positive" to give it cognitive significance.

Cheers.

- David

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1. Being isomorphic to the usual number system is not enough: two systems can be isomorphic without being conceptually similar. For instance, how would you rather deal with an object like a parabola-- as the graph of a quadratic function, or as intersections of certain planes with double cones, like Apollonius? Can you imagine doing trajectories being limited to such a framework? (Incidentally, how does this system of yours affect coordinate geometry?)

2. It is true that if you restrict the plane to the union of the positive x- and y-axes, the usual Euclidean inner product says that two nonzero numbers in that set are of "opposite sign" if and only if they are orthogonal. But in that case, you've essentially invented new notation for the negative sign that takes 3 to 4 times as long to write.

3. You gave your model a notion of positive the second you associated one of the slots in your ordered pair with 1 and the other with -1. Just because you don't use the word doesn't mean that isn't exactly what's going on.

In any case, we should probably return the thread back to its original topic. You can have the last word if you want.

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Have you read Ayn Rand's introduction to Objectivist Epistemology? In the expanded Second Edition this question is dealt with by Rand towards the end of the book [p304 of the Meridian published copy].

I'm a new newbie poster so I don't know if this will help, but here is the text:

Prof C: Some mathematicians claim that there is such a thing as an "imaginary number." How do you determine whether it is correct or not to include imaginary nubers within the same category as real numbers?

Ayn Rand: By defining the essential characteristics of the units. After you define what a real number is, if you see from what you mean by those terms that there are essential differences, then you can't include them in the same concept.

But this is really a question concerning theory-formation, not concept-formation. You are in the realm of epistemology of science. I will just say on this topic that you have to treat scientific concepts in exactly the same way, in principle, as you treat "table" and "chair." If somebody decided to put tales and chairs into one concept - on the grounds that you always see them together - but beds and automobiles in another because he can lie down in either, you would object to that. Why? Because you would say that he has organized his data by non-essentials. He has ignored essential similarities and essential differences abd arbitrarily coupled certain existents into certain groups. That error comes under the general category of definition by non-essentials. It is disastrous conceptually to try to integrate objects by non-essential characteristics.

She goes on to say [and this is the crux of the matter] -

If they serve that purpose [ie of measurement and method], then they have a valid meaning--only then they are not concepts of entities, they are concepts of method. If they have a use which you can apply to actual reality, but they do not correspond to any actual numbers, it is clearly a concept pertaining to method. It is an epistemological device to establish certain relationships. But then it has validity. All concepts of this kind are concepts of method and have to be clearly differentiated as such.

Material is copyrighted and reproduced for fair use.

Edited by UKObjectivist
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About your quote from Ayn Rand:

AR:If they serve that purpose [ie of measurement and method], then they have a valid meaning--only then they are not concepts of entities, they are concepts of method. ...

It is not exact that the purpose which AR was referring to was "of measurement and method", as you added in parenthesis: it was only method.

Indeed, Prof. C corrected the misunderstanding just a few lines above:

Prof. C: Excuse me, [these imaginary numbers serve] not in measurement of anything, but in computation - in solving an equation.

Thus, Ayn Rand's conclusion was that the imaginary numbers are pure concepts of method, not of reality; they don't measure anything real.

Alex

Edited by AlexL
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1. Being isomorphic to the usual number system is not enough: two systems can be isomorphic without being conceptually similar. For instance, how would you rather deal with an object like a parabola-- as the graph of a quadratic function, or as intersections of certain planes with double cones, like Apollonius? Can you imagine doing trajectories being limited to such a framework? (Incidentally, how does this system of yours affect coordinate geometry?)

Exactly my point. The traditional notion of integers obfuscates the fact that they are vectors. You haven't directly challenged this contention, can I assume you agree that integers are vectors, conceptually, even if that is not how they are traditionally formulated?

The system has the effect of changing the coordinate system via a linear transformation, and so changes nothing about the results obtained so far. It makes conceptualization of space no harder, whilst opening up new frontiers of conceptualization due to the ease with which certain hard traditional ideas can be grasped (such as relativity) and allows conceptualization of what I term time-space, a space defined by motion at constant speed, where direction of motion is the only control variable.

I know it sounds strange at first blush, but this is what is going on: every iota of existence is moving at the same speed, but each can change direction. A seemingly static object, like my coffee cup sitting on the table next to me, is actually a tiny little mindless bee buzzing around at super high frequency such that the after-image of its tracery is my cup. This is why gravity is essential, to keep the bees from wandering off in all directions indefinitely.

Speed is not a control variable. What we observe as translational motion is the average of the buzzing around, like a weighted random walk.

I'll be starting a new topic on the idea that instantaneous speed is not a control variable, i.e., that all identifiable items of existence, at every level of time-size relation, including photons, are moving at the same speed.

Interesting?

- David

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2. It is true that if you restrict the plane to the union of the positive x- and y-axes, the usual Euclidean inner product says that two nonzero numbers in that set are of "opposite sign" if and only if they are orthogonal. But in that case, you've essentially invented new notation for the negative sign that takes 3 to 4 times as long to write.

The conceptual leverage comes in two ways:

1. I get a means to represent time in addition to space, in the same framework, because the non-reduced pairs of natural numbers can be reduced to get the position, but also, the sum of the non-reduced components can be used to account the number of steps taken in reaching a location. In other words, (2,1) is the same spatial position as (1,0), but two steps later in time.

2. At higher orders, e.g., for planes, the efficiency really pays off because you have one less direction to worry about, and no negatives simplifies computations.

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3. You gave your model a notion of positive the second you associated one of the slots in your ordered pair with 1 and the other with -1. Just because you don't use the word doesn't mean that isn't exactly what's going on.

No I did not, you did. And you are wrong. I'm using plain old algebra here, not a special variant.

- David

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Magnitude is a multiple of a scalar unit.

Direction is towards or away from something, with respect to a focal point.

For example, in Euclidean 3-space, vectors have both magnitude and direction ... a vector may have any magnitude, and its direction correspond to one of the points on a sphere (the focal point is the origin).

- David

Edited by icosahedron
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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)?

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One doesn't need to count them, only realize that change involves both what kind of change (the direction), and how much of it (the magnitude).

Direction answers the question: What change?

Magnitude answers the question: How much change?

Magnitude can be, and often is, discrete, as in counting pennies. In such cases, the natural numbers can be used to account magnitude.

If I owe 100 pennies, I don't say that I own -100 pennies. Why does mathematics try to say that?

Thinking that negatives are something in and of themselves leads to conceptual difficulties, eventually, no less in mathematics than any other discipline.

Fact is, only the natural numbers are necessary to form rational numbers and hence obtain the ability to relate reality to units, and measure quantities. Real numbers are irrelevant to reality because the irrationals permeate the reals, and an irrational number cannot be the result of a real measurement.

If folk would stop with the demand for compactness, then the rationals would be a perfectly sufficient measurement space for quantities. But it is hard to do calculus without the (at least implicit) notion of compactness.

And if folk would create and maintain proper conceptual hierarchies based on personal verification, they wouldn't be tempted to conflate disparate concepts (such as scalar and vector).

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

- David

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