Nate Smith

Incommensurability & Irrational Numbers

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I have a friend who is working through some Aristotle in grad school, and he's dealing with the problem of incommensurability of numbers. Having more of a math education, he came to me with questions, but I don’t have this all figured out.

For those who haven't heard of this before, two numbers are incommensurable if they have no common unit of measure for which both numbers are whole number multiples of that unit.

Put another way, a and b are commensurable if there exists some number c such that a = mc and b = nc where m and n are whole numbers. They are incommensurable if there is no such number c.

For example, 23.516 and 4.1978 are commensurable. If we choose the number 0.0001 as our smallest unit of measurement, both of the numbers chosen are whole number multiples of that unit:

23.516 = 235160 x 0.0001

4.1978 = 41978 x 0.0001

Defined another way: if we choose any two lengths, and there exists some unit that measures both lengths evenly, the lengths are commensurable.

It may seem that any two lengths are commensurable, that there must be some length small enough to measure both evenly, but it was discovered in ancient Greece (by Pythagoras I have read) that this is not true. Take for example a square with sides of length one. Applying the Pythagorean theorem to obtain the diagonal, we have sqrt(2).

1 and sqrt(2) are incommensurable. There is no unit small enough that will measure each length evenly.

The same goes for pi. It is incommensurable with all whole numbers. Pi and sqrt(2) are called irrational numbers because, by definition, irrational number cannot be expressed as the fraction of two integers (the way that 3.5 can be expressed as 7/2).

Because they can’t be expressed as ratios of integers, all irrational numbers are incommensurable with all rational numbers. (This link has a good background on this topic.)

Now this raises some questions:

1) Does incommensurability create any metaphysical dilemmas?

2) Is it the case that mathematics at some level can’t match up with reality?

3) Do irrational numbers exist in reality?

Apparently this was a problem in mathematics for centuries.

According to my friend (and I may be misunderstanding him or Aristotle), Aristotle claimed that mathematical physics is not possible because of incommensurability (obviously this is presented out of context, but I’m just trying to raise some questions here).

I’d like to hear any thoughts on this for those aware of this problem. Thanks.

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1) Does incommensurability create any metaphysical dilemmas?

Mathematical, perhaps, but not metaphysical. A is A, regardless of what system we choose to measure its characteristics.

2) Is it the case that mathematics at some level can’t match up with reality?

What level are you referring to? We've sent shuttles to the moon and back on mathematics. I think that's fair proof that it matches up with reality.

3) Do irrational numbers exist in reality?

No number exists in reality, because numbers are abstractions. That does not mean that they are not objective; numbers measure real properties, and are created first by observing and relating quantities to each other.

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--------------

Apparently this was a problem in mathematics for centuries.

According to my friend (and I may be misunderstanding him or Aristotle), Aristotle claimed that mathematical physics is not possible because of incommensurability (obviously this is presented out of context, but I’m just trying to raise some questions here).

I’d like to hear any thoughts on this for those aware of this problem. Thanks.

Yes, it was a problem, which Ayn Rand solved in her book Introduction to Objectivist Epistemology. One cannot equate methodology with metaphysics.

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1) Does incommensurability create any metaphysical dilemmas?

Mathematical, perhaps, but not metaphysical. A is A, regardless of what system we choose to measure its characteristics.

In the event the question (and question #2) refers to an entity that is mathematically "infinitely" small/large, that has been discussed a few times on the Forum, and I am confused about how one can reasonably question the non-existence of a metaphysical dilemma.

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In the event the question (and question #2) refers to an entity that is mathematically "infinitely" small/large, that has been discussed a few times on the Forum, and I am confused about how one can reasonably question the non-existence of a metaphysical dilemma.

There aren't really any metaphysical dilemmas because reality just is what it is and A is always A. Dilemmas are always epistemological since someone has a dilemma or faces a conflict within their consciousness.

I think what you may mean is a dilemma concerning the nature of reality, i.e., a dilemma about metaphysics. In the case of infinity, the dilemma is resolved by explaining that infinity refers to a concept of method and not to a concept denoting an entity or existent.

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In the event the question (and question #2) refers to an entity that is mathematically "infinitely" small/large, that has been discussed a few times on the Forum, and I am confused about how one can reasonably question the non-existence of a metaphysical dilemma.

There aren't really any metaphysical dilemmas because reality just is what it is and A is always A. Dilemmas are always epistemological since someone has a dilemma or faces a conflict within their consciousness.

I think what you may mean is a dilemma concerning the nature of reality, i.e., a dilemma about metaphysics...

I think that by "reasonably question the non-existence of a metaphysical dilemma" she meant that she doesn't understand why anyone else would not by now understand that there is no problem, especially given how often this has already been discussed on the Forum.

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I have a friend who is working through some Aristotle in grad school, and he's dealing with the problem of incommensurability of numbers. Having more of a math education, he came to me with questions, but I don’t have this all figured out...

Now this raises some questions:

1) Does incommensurability create any metaphysical dilemmas?

2) Is it the case that mathematics at some level can’t match up with reality?

3) Do irrational numbers exist in reality?

Apparently this was a problem in mathematics for centuries.

According to my friend (and I may be misunderstanding him or Aristotle), Aristotle claimed that mathematical physics is not possible because of incommensurability (obviously this is presented out of context, but I’m just trying to raise some questions here).

I’d like to hear any thoughts on this for those aware of this problem. Thanks.

You should start with the threads where this has already been discussed. You started this one on infinite decimals yourself, and this one deals explicitly with incommensurability and sqrt(2).

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You should start with the threads where this has already been discussed. You started this one on infinite decimals yourself, and this one deals explicitly with incommensurability and sqrt(2).

All Interesting, and thanks for the links to other threads (I’ll read them when I get the chance.)

But I do have some questions and undeveloped thoughts about Metaphysics and its relationship to infinity:

Let’s say you video tape a ball bouncing across a room, it takes a total of five seconds and the camera is recording at 25 frames-per-second. At this rate the change from one frame to the next is clearly visible.

Here where things start getting weird; let’s say you increase the speed of the camera to 30 frames-per-second; there will still be change on each frame. How about 40, or 50, or 1000, or 1 million frames per second?

No matter how many times you divide up the 5 seconds it took that ball to bounce across the room, there would still have to be some change, even if it is at a practically unobservable (subatomic) level. Time is the measurement change in the relationship between and of objects; if what I mention above is true, then an infinite amount of change (time) would exist in a finite amount of space. This is highly counter-intuitive.

There are others examples that try to make this same point; the classic calculus example of the curve of a circle, or dividing up a pie; but those examples usually use abstract objects (like a perfect lined circle or triangle) or some other devise. I find this example most interesting because it deals with a very real mystery; it involves dividing up real change (time.)

Now, this is probably a mystery best solved by physicists, and not philosophers, but I am still interested.

I believe Betsy mentioned that infinity “refers to a concept of method and not to a concept denoting an entity or existent”; is what I mentioned above a method? It does require dividing up time (change), where no such division really exists; perhaps that is where this “dilemma” comes from, and this is just a case of being wrapped-up in some abstraction; Maybe measuring change (time) within another measurement of that same thing (time), is wrong on its feet.

If this has already been dealt with, sorry (but if you would, please point me in the direction it’s been talked about. :) ) Oh, and if anything above is fallacious or blatantly wrong in any way, I beg you to point it out.

- Ryan

A while ago, I started causally reading about the philosophy of mathematics (and its relationship to logic), which is not specifically dealt with in Objectivist literature.

I found a resource for further study, Foundations Study Guide: Philosophy of Mathematics.

Here are a couple of quotes, first from David Ross (Mathematician and Objectivist):

Objectivism recognizes a deeper connection between mathematics and philosophy than advocates of other philosophies have imagined. According to Ayn Rand's theory, the process of concept-formation involves the grasp of quantitative relationships among units and the omission of their specific measurements. It thus places mathematics at the core of human knowledge as a crucial element of the process of abstraction. This is a radical, new view of the role of mathematics in philosophy.

The next from Leonard Peikoff:

Mathematics is the substance of thought writ large, as the West has been told from Pythagoras to Bertrand Russell; it does provide a unique window into human nature. What the window reveals, however, is not the barren constructs of rationalistic tradition, but man's method of extrapolating from observed data to the total of the universe...not the mechanics of deduction, but of induction

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No matter how many times you divide up the 5 seconds it took that ball to bounce across the room, there would still have to be some change, even if it is at a practically unobservable (subatomic) level. Time is the measurement change in the relationship between and of objects; if what I mention above is true, then an infinite amount of change (time) would exist in a finite amount of space. This is highly counter-intuitive.

That is simply a variant of the classic "Zeno's Paradox". The paradox is that, before you can travel a length L, you first must need to travel L/2, and before that, L/4, etc. "ad infinitum". This is often used to illustrate the concept of limits which is important in calculus. In reality, if you take 1/2^n (2 to the power of positive integer n), and sum up all of the terms for n=1 through "infinity" (i.e. 1/2+1/4+1/8+...), the limiting value is very finite: just 1.

Whether *in fact and in reality* there are actually incommensurable units is a different question. Any real world measurement ultimately bumps up against the quantized nature of the real world, e.g. at a certain point your ruler is comprised of discrete atoms, and that's a point that happens long before 'n' in the above example reaches a mere 3 digits long.

I think it remains an interesting open question as to the exact way that mathematical abstractions relate to actual existence, though a few discussions with ewv, a very smart fellow who knows a ton of math, has convinced me that I need to think more about the subject before publicly posting a lot on it.

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No matter how many times you divide up the 5 seconds it took that ball to bounce across the room, there would still have to be some change, even if it is at a practically unobservable (subatomic) level. Time is the measurement change in the relationship between and of objects; if what I mention above is true, then an infinite amount of change (time) would exist in a finite amount of space. This is highly counter-intuitive.

The reason why is that you derailed at "an infinite amount of change." What does that mean? There is only a finite amount of change: from this position to that position, from this state to that state, etc.

I believe Betsy mentioned that infinity “refers to a concept of method and not to a concept denoting an entity or existent”; is what I mentioned above a method?

The process of continuing to divide things up is a method. You can mentally project what the result of that process might be but you can't, in fact, really continue to divide things up like that. Sooner or later you'll have to stop to eat lunch, etc.

It does require dividing up time (change), where no such division really exists; perhaps that is where this “dilemma” comes from, and this is just a case of being wrapped-up in some abstraction; Maybe measuring change (time) within another measurement of that same thing (time), is wrong on its feet.

Another problem I see here is that you seem to be assuming that time and change are the same thing. They are not.

Time is a measure of change -- the "how long it took-ness" of a change -- but there are others. For instance, distance is a measure of change of position -- the "how far it went-ness" of the change.

A change is a finite happening and time, distance, beginning-state/end-state, etc., are among the ways we measure it.

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Whether *in fact and in reality* there are actually incommensurable units is a different question. Any real world measurement ultimately bumps up against the quantized nature of the real world, e.g. at a certain point your ruler is comprised of discrete atoms[...]

An incommensurable unit of measurement dependent on the probability amplitude of a particle in a certain state in your tool of measurement (i.e. ruler)?

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Whether *in fact and in reality* there are actually incommensurable units is a different question. Any real world measurement ultimately bumps up against the quantized nature of the real world, e.g. at a certain point your ruler is comprised of discrete atoms[...]

An incommensurable unit of measurement dependent on the probability amplitude of a particle in a certain state in your tool of measurement (i.e. ruler)?

What does this mean?

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Whether *in fact and in reality* there are actually incommensurable units is a different question. Any real world measurement ultimately bumps up against the quantized nature of the real world, e.g. at a certain point your ruler is comprised of discrete atoms[...]

An incommensurable unit of measurement dependent on the probability amplitude of a particle in a certain state in your tool of measurement (i.e. ruler)?

No, he is saying that there are limits on the precision of measurement (though you would exceed the precision of an ordinary ruler long before reaching the atomic scale of Angstroms). But there are no metaphysically incommensurable lengths. That measurement is the conceptual common denominator that makes the concept "length" possible, although the actual numerical measurements are only implicit when forming the concept. Incommensurable would mean the lengths can't be compared as perceptually similar, so you couldn't group them under the same concept. (Length and red are incommensurable at the perceptual level.)

"Incommensurable" in mathematics is a technical term pertaining to the abstractions, not to metaphysics: it means there is no (finite) subdivision of the base unit possible which cannot be improved upon for a better precision numerical comparison. That is because the open-ended nature of concepts permits such an abstraction as irrational numbers with "infinite" decimal places, which in turn is required by the mathematical facts that give rise to it.

But you don't need a high degree of precision, let alone "infinite", to form the concept of length, which is done on the macroscopic level of human sense perception and which in turn becomes the context for all future methods of further refinement.

This is what Ayn Rand said at the epistemology workshops on "exactness" in measurement ("Appendix—Measurement, Unit, and Mathematics", pp. 194, 196):

With scientific development you might discover that, microscopically, the edge of this piece of paper is ragged and has tiny mountain peaks and valleys. That is not relevant to your [macroscopic] process of measurement, because you had to use the perceptual method as a start in order to get to your microscopic instruments of measurement...

Prof. E: Every measurement is made within certain specifiable limits of accuracy. There is no such thing as infinity in precision, because you are using some measuring instrument which is calibrated with certain smallest subdivisions. So therefore there is always a plus or minus, within the limits of accuracy of the instrument. And that's inherent in the fact that everything that exists has identity.Now if that's so, you can measure up to any specifiable degree of precision by an appropriately calibrated measuring rod.

If exactness in measurement is defined in such a way that you have to get the last decimal of an infinite series, by that definition no measurement can be exact. The concept of "exact measurement" as such becomes unknowable and meaningless, and therefore what would it mean to say a measurement is inexact? Exactness has to be specified in a human context, involving certain limits of accuracy. Is that valid?

AR: Yes, in a general way. But more than that, isn't there a very simple solution to the problem of accuracy? Which is this: let us say that you cannot go into infinity, but in the finite you can always be absolutely precise simply by saying, for instance: "Its length is no less than one millimeter and no more than two millimeters."

Prof. E: And that's perfectly exact.

AR: It's exact. If an issue of precision is involved, you can make it precise even in non-microscopic terms, even in terms of a plain ruler. You can define your length—that is, establish your measurement—with absolute precision.

She also recognized that the abstract concept sqrt(2) refers to the sequence 1.4, 1.41, 1.414,... etc.

Each referent (term in the sequence) of the concept, when squared, is equal to 2 within some precision: The square root of 2 is a number whose square is two within some precision, with what precision left unspecified as an omitted measurement when you form the concept. The concept is, like all concepts, open-ended, in this case with respect to the precision. A lot more can be said about sqrt(2), irrational numbers, and the mathematical continuum within the science of mathematics, but this is all anyone needs to know to form the concept sqrt(2).

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Whether *in fact and in reality* there are actually incommensurable units is a different question. Any real world measurement ultimately bumps up against the quantized nature of the real world, e.g. at a certain point your ruler is comprised of discrete atoms[...]

An incommensurable unit of measurement dependent on the probability amplitude of a particle in a certain state in your tool of measurement (i.e. ruler)?

No, he is saying that there are limits on the precision of measurement (though you would exceed the precision of an ordinary ruler long before reaching the atomic scale of Angstroms). But there are no metaphysically incommensurable lengths. That measurement is the conceptual common denominator that makes the concept "length" possible, although the actual numerical measurements are only implicit when forming the concept. Incommensurable would mean the lengths can't be compared as perceptually similar, so you couldn't group them under the same concept. (Length and red are incommensurable at the perceptual level.)[...]

Yes, my question was intended to point out how the bolded portion of PhilO's response could be interpreted because his response does not explicitly lay out the CCD as you do. Betsy, I trust this exchange addresses your question.

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I had in mind "incommensurable" in the mathematical sense of not being able to be expressed as the ratio of two integers, not in the sense of "a unit of length".

If the plenum of space is truly continuous, rather than at some level quantized itself, then logically sqrt(2), transcendental numbers such as Pi and e, etc., really do have exact metaphysical counterparts in terms of ratios of lengths. If it is at some level quantized, that would not be true. I think in some context it matters - the context of the application of mathematics in physics at the quantum level, especially in regards to QM and gravity. The working assumption when using mathematics predicated on the idea of true continuity is that reality is truly continuous "as far as you can go".

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If the plenum of space is truly continuous, rather than at some level quantized itself, then logically sqrt(2), transcendental numbers such as Pi and e, etc., really do have exact metaphysical counterparts in terms of ratios of lengths. If it is at some level quantized, that would not be true. I think in some context it matters - the context of the application of mathematics in physics at the quantum level, especially in regards to QM and gravity. The working assumption when using mathematics predicated on the idea of true continuity is that reality is truly continuous "as far as you can go".

Doesn't the idea of a “continuous” universe assume the existence of particles that are infinitely small? If that’s true, then the concept is itself invalid because it rejects Identity.

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Doesn't the idea of a “continuous” universe assume the existence of particles that are infinitely small? If that’s true, then the concept is itself invalid because it rejects Identity.

Not if particles are soliton waves in a continuous plenum - which combines discrete quantum nature with continuity at a more fundamental level.

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Not if particles are soliton waves in a continuous plenum - which combines discrete quantum nature with continuity at a more fundamental level.

What makes up the continuous plenum?

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What makes up the continuous plenum?

The continuous plenum. Reality isn't, and can't be, an infinite regress - it is, what it is. Continuity doesn't imply an infinite regress. It must have quite definite properties and identity; indeed, understanding them completely would theoretically permit the entirety of the laws of physics to be deduced, including things that are (in my current understanding) now considered more or less axiomatic such as inertia, and the properties of leptons (such as electrons), which are not now considered to have internal structure of any kind. No doubt there would also be new knowledge.

There is a hierarchy to reality that isn't "just epistemology". All of the normal matter that we observe out to the limits of observability, in all of its complex configurations, is ultimately comprised of less than 100 types of chemical elements, and protons, neutrons, and electrons make those up in turn, with the protons and neutrons ultimately comprised of quarks. Of course there are also photons, neutrinos, and some "exotic matter" in the mix. Particles have a discrete nature but they aren't "perfect points", for which there are various explanations (QM, Lewis Little's TEW theory, to take two overlapping ones).

A continuous plenum of the sort that I've envisioned abstractly would be at the "base" of everything, with the most elementary particles being solitons directly in the plenum, capable in turn of forming more complex configurations (i.e. the universe as we know it). Light would also be "waving" in the plenum as well; thus ultimately all interactions are at the level of a continuous plenum (e.g. matter-matter, matter-EM radiation, matter-antimatter, matter-in-motion, etc.) When I say "I've envisioned", I don't mean to imply that the general idea is mine, various forms of it have been around for a long time, though I may have some new ideas about it. I find the idea compelling because it provides a unified view of physics that also jibes with philosophy. For example, I don't find it satisfying to say that "somehow" an electron and positron change into two gamma rays after they collide and interact (i.e. the material, capable of travelling less than light speed, into photons travelling at light speed). There must be a way that actually happens. Einstein's e=mc^2 provides a quantitative explanation but not a structural one. etc.

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A small p.s.: there are more than 100 chemical elements possible, but only elements out to 92 (Uranium) are observed naturally in any significant abundance, hence my "less than 100" number in the prior post.

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What makes up the continuous plenum?

The continuous plenum. Reality isn't, and can't be, an infinite regress - it is, what it is. Continuity doesn't imply an infinite regress. It must have quite definite properties and identity;[...]

A continuous plenum of the sort that I've envisioned abstractly would be at the "base" of everything, with the most elementary particles being solitons directly in the plenum, capable in turn of forming more complex configurations (i.e. the universe as we know it). Light would also be "waving" in the plenum as well; thus ultimately all interactions are at the level of a continuous plenum (e.g. matter-matter, matter-EM radiation, matter-antimatter, matter-in-motion, etc.)

My understanding here is that the plenum in your current view does consist of discrete entities (in configurations) within that plenum. If the plenum consists of the interactions you posit then it cannot be physically continuous (uniform) in the same way that a solute cannot be dissolved and not dissolved in water simultaneously, nor can the same concentration of solute be distributed throughout the liquid evenly and simultaneously. Solitons still have boundaries and interactions between different solitons. I realize these are commonly-noted objections, but if you have some thoughts in rebuttal I would be interested. Of course, I could just be totally myopic and missing a new spin on this theory of everything.

If there is a metaphysical counterpart to a human's finite ability to count to mathematical infinity, even if by "ratios and lengths" you still require conceptual units created by a human to determine those ratios and lengths. It would just be an alternate measurement system of an objective reality that does not itself form conceptual units. Whether I say something is 3.141 or 3.141 leptons, I'm not seeing how numbers would no longer be a human concept of measurement but an attribute of the plenum. Working back up to perceivables, an infinity would at some level, be it individual soliton or any aspect of the plenum, be perceived by a human, a finite entity. I'm not seeing how a measurement system, with '0' or without a '0', can be both a measurement and a physical attribute. What am I missing?

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My understanding here is that the plenum in your current view does consist of discrete entities (in configurations) within that plenum. If the plenum consists of the interactions you posit then it cannot be physically continuous (uniform) in the same way that a solute cannot be dissolved and not dissolved in water simultaneously, nor can the same concentration of solute be distributed throughout the liquid evenly and simultaneously. Solitons still have boundaries and interactions between different solitons. I realize these are commonly-noted objections, but if you have some thoughts in rebuttal I would be interested.

Actually, macroscopic solitons (e.g. ones forming in water) simultaneously possess a discrete nature and a continuous nature. Not discrete as in "sharp boundaries", but as in certain regions of the medium that stably cohere together under movement and interactions within the continuous medium. Most waves disperse over time and do not have an entity-like characteristic. That's a fundamental appeal of the soliton-in-plenum hypothesis of the elementary particles, and it provides a natural solution to the purported (and of course impossible) contradiction in QM of wave vs. particle. No contradiction, a soliton is a particle-like special wave but with a smooth continuous integration into a continuous medium (well, in the macroscopic case, approximately continuous.) Here are some computer simulation/visualizations that I found, that demonstrate this idea with different classes of already known solitons:

http://www.math.h.kyoto-u.ac.jp/~takasaki/soliton-lab/gallery/solitons/kdv-e.html ://http://www.math.h.kyoto-u.ac.jp/~ta...ns/kdv-e.html (click on the gif for quickest viewing)

http://www-vis.lbl.gov/Vignettes/ (scroll down to "The nonlinear Schrodinger equation (NLS) ...")

http://www.sfu.ca/~renns/lbullets.html

http://www.crm.umontreal.ca/Galerie/gal_solitons.html

If there is a metaphysical counterpart to a human's finite ability to count to mathematical infinity, even if by "ratios and lengths" you still require conceptual units created by a human to determine those ratios and lengths.

I'm not sure what you mean by mathematical infinity in this context. Pi and e are of course finite numbers, but we can't "precisely write them out" in finite form; my point was simply that, at the level of the plenum, if it is continuous (and arguably must be), then it is in fact possible to have two distances within that plenum whose ratio is precisely e, Pi, or any other real number given an endless plenum. On the other hand, those particular numbers will never be precisely representable by any ratio of two sets of counted entities even if you have the entire known universe to play with.

Also, you can't have a non-integer number of entities (3.141 leptons, say :)); that's why I think that there's a real difference between integers and integer ratios, which relate to the counting of discrete things (entities), vs. ratios that arise from the relationship of attributes *of* things, which are continuous real numbers, i.e. integers relate to a higher level organization of the plenum in the form of entities, reals relate to relationships of attributes of those entities. Of course, you can see integers as merely a subset of the reals, but my point is that I think there's an implicit but real unit distinction between the two classes of numbers: integer problems, and continuous problems. For example, Diophantine equations are difficult equations solved in the context only of integers, typically (in my understanding) because the variables *do* represent the counting of discrete entities; you can't have 1/2 of a dime or a penny when you're finding the possible ways to make change for a specific amount. Pretending that the variables can just as well be reals won't get one far in solving them.

Part of my point is that these relationships have a metaphysical basis, and trying to elucidate exactly and explicitly how mathematical concepts relate to reality. Of course it takes human brains to conceptually identify those relationships, but e.g. the particular angle formed in a water molecule between the oxygen and the hydrogens is from a human-independent metaphysical reality, and it is also true that chemical reactions actually occur on an integer basis because entities (atoms) react with other entities, which is why stoichiometric chemical equations have to be balanced with integer coefficients to be realistic.

I don't want to make too much of the mathematical ideas here, I still have a lot of thinking to do.

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Actually, macroscopic solitons (e.g. ones forming in water) simultaneously possess a discrete nature and a continuous nature. Not discrete as in "sharp boundaries", but as in certain regions of the medium that stably cohere together under movement and interactions within the continuous medium. Most waves disperse over time and do not have an entity-like characteristic. That's a fundamental appeal of the soliton-in-plenum hypothesis of the elementary particles, and it provides a natural solution to the purported (and of course impossible) contradiction in QM of wave vs. particle. No contradiction, a soliton is a particle-like special wave but with a smooth continuous integration into a continuous medium (well, in the macroscopic case, approximately continuous.)

Thanks for the links but I'm not seeing how that responds to my concern. The characteristics of a soliton of the amplitude decreasing post-collision/interaction and subsequent soliton spreading are finitely measureable characteristics of an entity regardless of the fact that overall there is continuous integration. A soliton-as-and-in-plenum hypothesis takes care of the wave vs. particle problem, but again, solitons are finitely measureable components. Multidrug delivery-in-solitons would not be otherwise be possible.

If [the plenum] is continuous (and arguably must be), then it is in fact possible to have two distances within that plenum whose ratio is precisely e, Pi, or any other real number given an endless plenum.

Mathematically, yes, but all I am saying is that you could not ever physically measure that.

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Thanks for the links but I'm not seeing how that responds to my concern.

Sorry, I'll try again. I missed what I think is an important part of your original objection:

My understanding here is that the plenum in your current view does consist of discrete entities (in configurations) within that plenum. If the plenum consists of the interactions you posit then it cannot be physically continuous (uniform) in the same way that a solute cannot be dissolved and not dissolved in water simultaneously, nor can the same concentration of solute be distributed throughout the liquid evenly and simultaneously.

Continuous is not meant to imply that the medium has a uniform state throughout, but that, in some sense, there are no "gaps" in the plenum, no "regions of literal nothingness", which is philosophically objectionable (and which would logically exist if existence were ultimately a bunch of little particles with hard boundaries.) But clearly, it would be a boring world if nothing could change, so this continuous medium must be able to take on different state conditions at different regions in space.

To take more water examples: the presence of waves (solitonic or not), of currents, of *ice floating*, in a body of fresh water (to remove the possible confusion of sea salt in this example), does not imply that there is anything but water there. It's water, but it's water in a certain state: some parts are moving at different velocities, some parts are in a solid phase, some in a liquid phase - but it's still *water*. A solitonic wave moving through the water carries energy and has a particular identity and is entity-like, but it's really just a configuration of water-within-water and it is certainly smoothly integrated with the rest of the body of water. All of these phenomena are known to exist.

So, the big question is how a continuous plenum can be continuous but have spatio-temporal state changes - a sort-of equation of state. Exactly how states evolve to other states would be the root of all causality. The water example is just a crude analogy (the plenum is certainly not material, matter is an emergent property in this idea) but a useful one; there are no mysterious empty regions in the water in order for internal changes to occur in what is, to a significant approximation, an incompressible continuous medium.

I hope that helps to answers you.

If [the plenum] is continuous (and arguably must be), then it is in fact possible to have two distances within that plenum whose ratio is precisely e, Pi, or any other real number given an endless plenum.
Mathematically, yes, but all I am saying is that you could not ever physically measure that.

I'm not sure what you mean by this.

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An energy packet moving at 135 knots is really just a configuration of sustained internal winds and interacts with winds surrounding it. It has a particular identity and it is an entity known as a hurricane. Definitely solitons possess identity; I'm not sure why it can't be defined as an entity in the non-macroscopic arena (I have only thought about very Earthly solitons while using CFD software that examines the behaviour of solitons in "meshes", and yes, I did read what you wrote), and as long as they are not, solitons can be used in a ToE hypothesis as you have.

If [the plenum] is continuous (and arguably must be), then it is in fact possible to have two distances within that plenum whose ratio is precisely e, Pi, or any other real number given an endless plenum.
Mathematically, yes, but all I am saying is that you could not ever physically measure that.

I'm not sure what you mean by this.

See your own remark:

On the other hand, those particular numbers will never be precisely representable by any ratio of two sets of counted entities even if you have the entire known universe to play with.
Part of my point is that these relationships have a metaphysical basis, and trying to elucidate exactly and explicitly how mathematical concepts relate to reality.

Have you read Galois?

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