Numbers: A Physical Handshake with Design

ucarr

This OP keys off Philosophim’s conversation: A first cause is logically necessary.

There must be a starting-point physical entity, whether differentiable, or not. So, there too must be a starting-point counting number.

There’s no reductio ad absurdum re: two simultaneous starting-point things because these two starting point things are bi-conditionally connected. They are one-and-the same in the sense that they are essential attributes of each other. Where there is thing there is number, and vice versa.

These paired starting-points, operating in spacetime beginning-less, unbounded, finite and relativistic, exemplify the necessary arbitrariness of pre-analytic beginnings.

Since any and all material objects, individually, present as a countable one, oneness, a countable number, acts as an essential attribute of each and every material object.

When we postulate any object, we necessarily, bi-conditionally postulate a number one. Material object and number cannot be divorced.

Number is an essential, material property.

Moreover, given the multiplicity of solo ones, we must conclude all material objects are countable, both individually and collectively.

Therefore , material objects can always be gathered together into collections, that is sets.

Solo ones and sets entail physically real numbers as one of their essential, physical attributes. That we never fail to differentiate differently numbered sets, neither perceptually nor conceptually, evidences the essential status of number, the physical property.

Herein, there is no conflation of sign with referent because of the number zero. It is an absentially real number whose presence changes other numbers in its field of influence. As the mind, another, absentially real, presence, in parallel physically changes material objects within its field of influence.

ucarr

This is a continuation of the first part of my OP.

Conclusion: number-as-property, being essential and physically real, and being tied inextricably to material objects, is discovered. They are not purely conceptual objects, accessible to the mind only.

Numbers are discovered, not invented. Numerical properties and numerical relationships likewise are discovered, not invented.

The number zero shows how emptiness is permeated by these same numerical relationships, so existence presupposes number-as-property.

If number-as-property is physical and essential, then there is an answer to an important question: How can mental “objects” have causal effects upon the physics of the natural world? The answer is numbers.

This is the necessary conclusion of both the number-idealist and the number-realist.

Example: Civil engineering demonstrates idealist control of the physical if you’re a number idealist: The building of a suspension bridge across a body of water as, say, San Francisco’s Golden State Bridge, demonstrates numbers manipulated to specifications required for a stable road across a bay. How ideal number shakes hand with real object remains to be explained.

If, on the other hand, you’re a number-realist, then you understand there’s no unexplainable interface of ideal and real in the design and build of a suspension bridge. Numbers, like the bridge itself, are physically_materially real. The two are integrated and holistically consistent.

tim wood

There must be a starting-point physical entity, whether differentiable, or not. So, there too must be a starting-point counting number. — ucarr

"Must be"? Why must there be? If you look closely enough, you will find the imperative securely rooted in your need for one, in the (your, and mine too) logic of the thing. But logic is descriptive and only seems to be prescriptive. That, or show, extra-logic, how and why it must be.

ucarr

"Must be"? Why must there be? If you look closely enough, you will find the imperative securely rooted in your need for one, in the (your, and mine too) logic of the thing. But logic is descriptive and only seems to be prescriptive. That, or show, extra-logic, how and why it must be. — tim wood

The idea here is that with any and all experiences of sentient existence, the sentient being must make a start, i.e., embark upon their personal history. Making a start and making a starting count are one and the same. This isn’t the logic of the starting; there can be no logic of the starting as there is, as yet, no logic. Starting is pre-analytic, thus pre-logical. Starting with an arbitrary start_starting count is an existential necessity that has no logical support. This is evidenced by the scientific method: science starts with an arbitrary starting point, the axiom.

JuanZu

↪ucarr

If I have 5 oranges in one basket and I have 5 apples in another basket, the 5 does not seem to participate in Appleness nor the orangeness. So the number is not the same as numbered things. "5" is not oranges, nor apples, it only applies extrinsically. Therefore is wrong to say that "5" is physical "because apples and oranges are physical". You must say that numbers are physical things by itself to the extent of numbered things (apples and oranges).

Then it is necessary to define what you mean by a physical thing.

tim wood

This isn’t the logic of the starting; there can be no logic of the starting as there is, as yet, no logic. Starting is pre-analytic, thus pre-logical. Starting with an arbitrary start_starting count is an existential necessity that has no logical support. This is evidenced by the scientific method: science starts with an arbitrary starting point, the axiom. — ucarr

Amen, amen, & again amen. I wonder what fish will nibble here and be caught.

Hallucinogen

The structure you are looking for is an identity - a mathematical space/expression/set of operations that is always true.
Counting numbers originate from the fact that the identity self-distributes its own Boolean algebra. The set in its entirety (unity) corresponds to "1" and the empty set to "0". Subsetting allows the construction of von Neumann ordinals - sets that correspond to counting numbers.
Because physics consists of a set of points with trajectories in a mathematical space, this structure is everywhere-distributed in physics. That explains the connection between your "starting-point physical entity" and your "starting-point counting number". They do not "begin" simultaneously though - every physical fact depends on facts about this mathematical structure, but not vice versa.

180 Proof

From an old thread ...

My 'anti-platonist pragmatics' (finitism?) comes to this: pure mathematics is mostly 'invented' (re: pattern-making) and applied mathematics is mostly 'discovered' (re: pattern-matching). — 180 Proof

Like the rules and strategies of (e.g.) chess, respectively (i.e. grammars and narratives).

ucarr

f I have 5 oranges in one basket and I have 5 apples in another basket, the 5 does not seem to participate in Appleness nor the orangeness. So the number is not the same as numbered things. — JuanZu

You say number stands apart from apples and oranges . When we look at number five apart from them, we know nothing about their number. How do you know both have number five?

If it were the same (or if the number is an intrinsic property of numbered things), we would have to say that 5 apples are 5 oranges and vice-versa (or that 5 apples have the property of been 5 oranges and vice-versa) breaking the identity principle. — “JuanZu

Since number five, in abstraction, tells us nothing about apples, oranges or any other physically real thing, that tells us pure math, in order to be physically real and thus inhere within particular, physical things, and thus be existentially significant, meaningful and useful, must evaluate down to physical particulars. Universals are emergent from particulars, but they are not existentially meaningful in abstraction.

ucarr

If apples and oranges have intrinsic physical properties then the number ~~(if it is different from numbered things to avoid breaking with the principle of identity)~~ does not participate in those physical intrisic properties either. Therefore, the number is not something physical and is extrinsic to intrinsic physical things which are numbered — JuanZu

Assuming you possess proper vision, have you ever been unable to distinguish five oranges from two oranges?

It is also necessary to define what you mean by a physical thing. — JuanZu

Physical: anything subject to the spacetime warpage of gravitational fields.

ucarr

…every physical fact depends on facts about this mathematical structure, but not vice versa. — Hallucinogen

So, pure math includes relationships without reference to physical things inhabiting the natural world. This is an intriguing argument for idealism. What do math theoreticians say about the physical mind’s ability to cognize these supposed ideals-in-themselves?

Do you believe math is metaphysically prior to physics? If so, what say you about the fact that math, like physics, possesses pre-analytical axioms? (They’re solely existential.). Also, what say you about math axioms being incomplete? (If they’re incomplete, they’re not ideals.)

ucarr

My 'anti-platonist pragmatics' (finitism?) comes to this: pure mathematics is mostly invented (re: pattern-making) and applied mathematics is mostly discovered (re: pattern-matching) — 180 Proof

Pattern-making in total abstraction from physical reference, beyond convention established by precedent, tells you what?

JuanZu

You say number stands apart from apples and oranges . When we look at number five apart from them, we know nothing about their number. How do you know both have number five? — ucarr

We do not know. "Have" refers to property. I prefer to say apples and oranges can be counted as something potential. We learn to count. But first things appear differentiated. The number is also differentiated, and in that case we would speak of isomorphism.

Since number five, in abstraction, tells us nothing about apples, oranges or any other physically real thing, that tells us pure math, in order to be physically real and thus inhere within particular, physical things, and thus be existentially significant, meaningful and useful, must evaluate down to physical particulars. Universals are emergent from particulars, but they are not existentially meaningful in abstraction. — ucarr

Well, they always tell us something significant and meaningful about themselves. Whether they are useful or not would be something extrinsic.

Physical: anything subject to the spacetime warpage of gravitational fields — ucarr

How is number 5 deformed by gravitational fields

180 Proof

↪ucarr

I don't understand the question. Rephrase?

ucarr

↪JuanZu

We do not know. — JuanZu

If I have 5 oranges in one basket and I have 5 apples in another basket… — JuanZu

If this is something you cannot know, then your argument above has no grounding in fact and therefore no logically attainable truth content, only blind guesswork. On that basis, why should I accept it?

JuanZu

If this is something you cannot know — ucarr

I have said "I have" as I can say "I have counted." To you it seems that apples and oranges are numbers, to me their numeration may simply be external properties that are only acquired in relationship. The latter is true and the former is false (due to impossibility to identify the number and the numbered).

then your argument above has no grounding in fact and therefore no logically attainable truth content, only blind guesswork. On that basis, why should I accept it? — ucarr

Because I have also indicated something that I do know: That the apples and oranges have been counted and have been subjected to number. And because they are differentiated (they are in a different place, for example, or do not share the same space) they can be counted.

Hallucinogen

Do you believe math is metaphysically prior to physics? — ucarr

It has to be, since mathematical concepts are more general than physical entities, which only exist at a given coordinate in space. Mathematical truths whoever enjoy far greater comprehensivity.

What do math theoreticians say about the physical mind’s — ucarr

I don't presuppose the existence of "physical minds".

If so, what say you about the fact that math, like physics, possesses pre-analytical axioms? — ucarr

What a priori axioms does physics possess? Any that math possesses supports my position.

Also, what say you about math axioms being incomplete? — ucarr

They are more general than physical rules but less general than logic and also less general than the global identity.

ucarr

↪180 Proof

Suppose the Riemann hypothesis finds its solution in pure math. So, pure math establishes that all primes calculable by the zeta function locate themselves on the critical line of the complex number plane.

Now let’s blink out the natural world of physics, thus leaving us with pure math with no physical referents, no matter how far down the line you evaluate. What are we left with? A system of interrelated signs with meaning and use resting upon nothing but the conventions implied by the system of signs itself, as established by the precedent, again, of the system itself.

What do we have? An endless loop of circular reasoning with no other meaning than its circularity. That’s why I say math is a physical property of the natural world. Only there does number possess existentiality, meaning and usefulness.

tim wood

It has to be, since mathematical concepts are more general than physical entities, which only exist at a given coordinate in space. Mathematical truths whoever enjoy far greater comprehensivity. — Hallucinogen

Hmm. I believe, subject to correction, the question is which is prior. To my way of thinking, any concept requires a mind to conceive it. Now you may very well say that minds seem to be pretty good at modelling and describing the world, but what does the world care about minds, models, and descriptions? It operated before there were minds - no minds required, thus no concepts required.

A particle moves through space; some formulae do a good job of describing that movement and even predicting how it might go. But the particle and its movement are clearly prior. Mathematics, then, would seem to be derive from the world, the world in every sense prior.

Wayfarer

Number is an essential, material property. — ucarr

Why 'material'? In what sense? In what sense is pure maths concerned with physical objects?

Now let’s blink out the natural world of physics... — ucarr

Rather an odd expression, but surely one of the confounding things about mathematics, is its applicability to physics. That is the basis of Eugene Wigner's celebrated essay, The Unreasonable Effectiveness of Mathematics in the Natural Sciences. I will not propose to explain this 'unreasonable effectiveness', as Wigner himself could not, and he a Nobel-prize winning mathematical physicist. But I would defend the modest claim that one of the grounds for the great successes of modern mathematical physics, was the discovery of the means by which to express the measurable attributes of physical bodies using mathematical logic. This enabled for great predictive success, whereby predictions are made that appear to 'fall out of the equations', but which lead to real-world discoveries such as Dirac's discovery of anti-matter. (And think about the etymology of the word 'discovery', for that matter.)

You would think that if mathematics were purely conventional, it would lack this ability to make genuine, unexpected discoveries about nature. The surprising effectiveness of mathematics in making accurate, sometimes unexpected predictions about the natural world suggests a deeper connection between mathematical structures and physical reality. This view opposes the idea that mathematics is just a tool invented for practical purposes, instead hinting at some intrinsic relationship between mathematical concepts and the fabric of the universe.

But in all this, I fail to see why we should accept that numbers are material properties. They may be applied to the measurable attributes of material bodies, but that is completely different to saying that they are material entities.

Tom Storm

he surprising effectiveness of mathematics in making accurate, sometimes unexpected predictions about the natural world suggests a deeper connection between mathematical structures and physical reality. This view opposes the idea that mathematics is just a tool invented for practical purposes, instead hinting at some intrinsic relationship between mathematical concepts and the fabric of the universe. — Wayfarer

Could it be that maths, like space and time are part of our human cognitive apparatus in some way?

ucarr

↪Hallucinogen

It has to be, since mathematical concepts are more general than physical entities, which only exist at a given coordinate in space. Mathematical truths whoever enjoy far greater comprehensivity. — Hallucinogen

Quantum computing has something contrary to say about the last part of your claim.

I don't presuppose the existence of "physical minds" — Hallucinogen

If mind emerges from brain, then no brain, no mind. Yes, mind is independent of brain as mind, but the absential materialism of mind is its constraints upon dynamical, material processes. Again, no dynamical, material processes, no mind. Functional mind that has impact upon existentiality, meaning and usefulness is never uncoupled from the physicality of the natural world.

What a priori axioms does physics possess? — Hallucinogen

What a priori reason is practiced by brain in a vat never in contact with the world?

Any that math possesses supports my position. — Hallucinogen

Math, like brain in a vat without the worldly mediation of conscious human, can only instantiate the circularity of thing-in-itself, and that without cognition. Pure math has connection to the natural world only as indecipherable signification representing thermodynamic equilibrium.

Since mathematicians only use pure math for investigation of the ground rules concerning applied math, pure math is merely higher-order applied math and thus it is not uncoupled from the natural world.

ucarr

In what sense is pure maths concerned with physical objects?
28 minutes ago — Wayfarer

Since mathematicians only use pure math for investigation of the ground rules concerning applied math, pure math is merely higher-order applied math and thus it is not uncoupled from the natural world. — ucarr

ucarr

A particle moves through space; some formulae do a good job of describing that movement and even predicting how it might go. But the particle and its movement are clearly prior. Mathematics, then, would seem to be derive from the world, the world in every sense prior. — tim wood

:up:

Speak! Without antecedent, existential fact, science and math can’t even get started.

Wayfarer

Now let’s blink out the natural world of physics, thus leaving us with pure math with no physical referents, no matter how far down the line you evaluate. What are we left with? — ucarr

Something very like Kant's 'concepts without percepts are empty'. 'Not uncoupled from the material world' does not mean 'material in nature'. Humans are metaphysical beings - we can peer into the intelligible domain and return with things like computers and airplanes.

jgill

I hesitate to enter this conversation, but I can't resist offering a few opinions held by an actual mathematician who has done both teaching and research.

The distinction between pure and applied math is somewhat vague, one reason being that pure math may become applied math at times. A researcher in applied math could be working on a math scheme to solve a particular problem, like calculating the stresses on a modern fighter plane during sharp turns. Or, he could be pursuing a topic purely for its own sake, curious about what comes next - and then finds someone has used his results in an applied manner.

This happened to me. My interests are always in "pure" math (complex analysis) and I published a paper in 1991, I think, with no thoughts of it ever being "useful", only to find my principle result was employed in a multiple author sociology paper about decision making in a group. Of course, the author who cited and used my result paid no attention to the details.

There are so many kinds of mathematicians and so many kinds of mathematics it's foolish to try to generalize. I'm guessing about 1200 PhDs are granted each year in the USA, with about half being American citizens. Probably a large majority never publish more than, say, two papers in their careers - for various reasons. But there are those, like myself, that find the exploration of new ideas fascinating.

That's how I see myself and many others: Explorers. It's no wonder you find mathematicians among rock climbers and mountaineers.

Pure math has connection to the natural world only as indecipherable signification representing thermodynamic equilibrium.
Since mathematicians only use pure math for investigation of the ground rules concerning applied math, pure math is merely higher-order applied math. — ucarr

Mysteries never cease :roll:

180 Proof

↪jgill

I appreciate your insights. :up:

jgill

The surprising effectiveness of mathematics in making accurate, sometimes unexpected predictions about the natural world suggests a deeper connection between mathematical structures and physical reality. This view opposes the idea that mathematics is just a tool invented for practical purposes, instead hinting at some intrinsic relationship between mathematical concepts and the fabric of the universe. — Wayfarer

Could be. :up:

ucarr

Not uncoupled from the material world' does not mean 'material in nature' — Wayfarer

Percept + concept = complex materialism. As with complex numbers, there is a real part and another type of part. In the case of materialism, the other type of part is non-local materialism: collection across time-interval-positive of a set of conditionally connected members reified into a gestalt, with gestalt in this context being a synonym for concept.

tim wood

I hesitate to enter this conversation, — jgill

And now the hubris of the one who knows will be punished by them what don't. But just a question: some folks think math invented, and others math discovered in the sense of its being "out there" somewhere. My view is that it is discovered, but in the only place it can be found, in a mind and not "out there." And thus discovered/invented, together. What do you say, if you care to say?

Wayfarer

Percept + concept = complex materialism. — ucarr

Tosh. Kant detested materialism, as do I.

Another mathematician, but one who ventured into philosophy and cognitive science, was Charles S. Pinter (whom I discussed briefly with jgill in the past (although I learn Pinter has since died, but then, he was 96 at time of death)). Anyway, his maths books are here, about which I know nothing, but his final work was the very interesting Mind and the Cosmic Order, published in Feb 2021, a
detailed abstract of which can be found here, and to which I would look for a possible answer to this question:

Could it be that maths, like space and time are part of our human cognitive apparatus in some way? — Tom Storm

Numbers: A Physical Handshake with Design

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