Closer to Truth: Is Mathematics Eternal?

I think it's easier to perhaps consider the Eternal realism of that which underlies maths, the very instruments of logic like Modus Ponens.

Admittedly I am very close to the Platonists, if not one myself where reality of logical/mathematical Universals is concerned, so might be biased.

Anyone happen to watch that BBC series about whether math is invented or discovered presented by hot ginger mathemetician boffin, Hannah Fry?

No one knows. Some maths seem to apparently have been invented, like '0' or 'infinity'. Others appear to be of natural law and therefore possibly discovered.

I read one of them books on physics by one of them popularizer physicists. The question was posited: 'Can a civilization create technology (or science?) without using mathematics'  (or to paraphrase a bit)

Surprisingly, the answer was yes!  Don't remember the explanation as to how though, lol.

I always wondered that. And out of the myriad of physics books I've read in my life, that was the first book that actually asked that question.

Good points, everyone... however, our current systems are peculiar to us. We created them in order to make sense of the measurable world around us.

The concepts of mathematics themselves arose from consciousness in the first place ~ mathematics makes no sense when it comes to the unmeasurable, however, like emotions, ethics, painting, music, meditating, thoughts, etc, etc. Majority of the time, we don't even think mathematically.

Does simply counting things count as mathematics? It may be a basis of mathematics, but I've always conceptualized it as a higher art based on abstract thinking beyond mere counting of objects or measuring things relative to other another, without numbers.

Feel free to correct any possible misconceptions.

Please note that I'm seeking to belittle mathematics or its usefulness, just that I want to highlight that I don't its some mystical or divine concept.

Maybe there is a divine mathematics, even, but I don't think it would be anything we'd recognize with our egos that are constricted to a narrow understanding of reality based on physical senses.

Poster kairosfocus at Uncommon Descent says it better than I could (at https://uncommondescent.com/philosophy/l...f-reality/ ). Some excerpts:

"Now, to the mathematical substance of reality. We see from distinct identity of a world, that the natural numbers are necessarily substantially present given the force of that identity. But, isn’t all of this just in our heads, if we aren’t here to do math it vanishes for want of sufficiently complex brains, poof.

Nope.

|| + ||| –> |||||

obtains whether we are there to perceive and contemplate it or not. Two is even and the first prime. Three is prime simply as it cannot be evenly shared in a whole number of slots apart from by ones. And so forth, property after property. We come along, discover significance, label, embed in systems of thought. But all of that is in response to substance that we find it necessary to accurately describe.

We develop symbols:

2 + 3 = 5

Those are cultural, convenient, helpful. But they are not the substance, they are our way to handle that substance. A substance of reality that is quantitative (amenable to measure) and structural (bound up in coherent relationships). We label such, Mathematics. That does not create a discipline by the poof-magic of words, it is a way for us to refer to the substantial reality and to our explorations and development.

What is space? We see here a quantitative structure tied to length and dimensionality. There is more or less of length and it can be in different directions. To get there conceptually, we need integers (thus negative numbers), we need fractions, we need power series that can sum to infinite numbers of terms and which converge in some cases, as we discovered — nope, it was not an invention — the irrationals.

Likewise, 0 = 1 + e^i*pi should not only give high confidence in the coherence of vast domains in maths, but it should give us a view into the roots of reality.

...to get to the substance of the gravitation that is at root of the force that triggers the accelerated motion, we find not only further structures and quantities such as displacement x, velocity dx/dt and acceleration dv/dt as well as the panoply of infinitesimals, limits etc, but also the warping of spacetime through the influence of a massive body, a planet. (A ship or a mountain shows similar effects, on a smaller scale.)"

The hierarchy of ontology and the foundation of mathematics could look like this:

Nothing
Intention
Experience A - Prime memory
Change
Experience A(B): Experience B nested within Experience A
     or with the memory  of A (this implies an experience of identity or oneness nested within the experience of difference)
Dimensions: Similarity / Difference spectrum emerges
    Identity: the "=" sign or the "1"
    Difference: 1 compared to 0
Pattern discovery: digitizing the analog - arbitrarily assigning a similarity threshold to consider Experience N identical to or not identical to Experience N(B)
       the assignation of the similarity threshold is based on intention, purpose, and choice
Pattern overlay: applying the identical arbitrary similarity threshold to find more identities in nested Experience N(C), N(D) N(E), ... etc.
Metaphor: a thread of structure connecting similar differences and different similars
Symbol: a condensed form for a metaphor
The Platonic realm (or as I like to call it, the 5th dimension): the entire symbolic metaphorical structure of all of reality - a map of all pattern overlays 
The Physical realm: an instantiation of the Platonic realm - a symbolic representation of experience - present experience cast into form
Mathematics: a symbolic representation of the Platonic realm.
Physics: Math applied to the physical realm.

Some far-out speculation. It occurs to me that even though the roots of the nature of the universe seem to necessarily follow the laws of logic and mathematics (which are a discovered truth by man), is there a possible reality in which this would not be the case? In this perhaps possible reality these roots would be merely a choice that was made by higher beings. 

In our world 2 + 2 might actually = 5. This would be if our universe were a virtual reality simulation and the rules by which it operates are merely choices made by the "programmer(s)". In this scenario, suppose that in their higher reality they instead decided to deliberately make our world irrational by our current thinking. Of course the workings of our minds would also have to be formed this way so that to us this would seem perfectly rational. But the laws of logic and mathematics would be vastly different than what we presently consider to be the absolutely necessary roots of reality.

Some far-out speculation. It occurs to me that even though the roots of the nature of the universe seem to necessarily follow the laws of logic and mathematics (which are a discovered truth by man), is there a possible reality in which this would not be the case? In this perhaps possible reality these roots would be merely a choice that was made by higher beings. 

In our world 2 + 2 might actually = 5. This would be if our universe were a virtual reality simulation and the rules by which it operates are merely choices made by the "programmer(s)". In this scenario, suppose that in their higher reality they instead decided to deliberately make our world irrational by our current thinking. Of course the workings of our minds would also have to be formed this way so that to us this would seem perfectly rational. But the laws of logic and mathematics would be vastly different than what we presently consider to be the absolutely necessary roots of reality.

Well if in the VR world, the rules of arithmetic were all normal except for 2+2=5, people would soon discover. You would gather apples one at a time - 1,2,3,4. Then someone would discover that it was better to set two people on the task, each would collect 2, and then combine them and get 5. Then someone else would watch the apples on the ground, and observe one appearing from nowhere.

Thus maybe even in a VR world mathematics could only be different by some sort of renaming (which isn't really a difference at all) such as using the symbol 5 to mean 4 and also using the symbol 4 to mean 5!

What can Music tell us about the Nature of the Mind? A Platonic Model

Brian Josephson & Tethys Carpenter

We present an account of the phenomenon of music based upon the hypothesis that there is a close parallel between the mechanics of life and the mechanics of mind, a key factor in the correspondence proposed being the existence of close parallels between the concepts of gene and musical idea. The hypothesis accounts for the specificity, complexity, functionality and apparent arbitrariness of musical structures. An implication of the model is that music should be seen as a phenomenon of transcendental character, involving aspects of mind as yet unstudied by conventional science

Elsewhere (Josephson and Carpenter 1994) the authors have commented on the existence of interesting parallels (principally involving matters concerning information and regulation) between aesthetic processes and life processes. These parallels will now be developed further.

In the present context, the most basic parallel is that between effective musical pattern and gene, both being informational structures playing a key role in the activities of those structures that contain them (organism and musical mind). In the case of life, the genes help to determine the forms and activities of the structures that cause the genes to be replicated so that they survive. Particular gene structures generate particularly effective functional systems, and this very often entails high complexity since complex means are generally needed to produce simple results in an effective manner. Other contributions to complexity come from the complexity of the chemistry involved and the fact that genes often do not produce their effects in isolation. Further, in organisms there are clear means-end relationships related to the functionality of structures, by virtue of which the functional structures can be considered 'significant'. In contrast to the perspicuity of the
processes involved at the functional level, the details at the structural level are complex, and related to function in a complex way, which generally has arbitrary aspects as a consequence of the way that nature operates opportunistically rather than logically.

We now observe the ways in which music possesses features paralleling those discussed in the case of life:

(i) effective musical structures are highly specific, as well as being (subjectively) functional;

(ii) while there is an overall logic behind the way that a given piece of music works, many of the details of form appear essentially arbitrary. The functional descriptions are considerably simpler than descriptions at the detailed level of the structure.

I don't know enough about composition to say much, but a lot of Josephson's speculations are at the least interesting...

Following the above which was 50% Josephson, another physicist - this time it's Wigner:

The Unreasonable Effectiveness of Mathematics in the Natu

THERE IS A story about two friends, who were classmates in high school, talking about their jobs. One of them became a statistician and was working on population trends. He showed a reprint to his former classmate. The reprint started, as usual, with the Gaussian distribution and the statistician explained to his former classmate the meaning of the symbols for the actual population, for the average population, and so on. His classmate was a bit incredulous and was not quite sure whether the statistician was pulling his leg. "How can you know that?" was his query. "And what is this symbol here?" "Oh," said the statistician, "this is pi." "What is that?" "The ratio of the circumference of the circle to its diameter." "Well, now you are pushing your joke too far," said the classmate, "surely the population has nothing to do with the circumference of the circle."

Naturally, we are inclined to smile about the simplicity of the classmate's approach. Nevertheless, when I heard this story, I had to admit to an eerie feeling because, surely, the reaction of the classmate betrayed only plain common sense. I was even more confused when, not many days later, someone came to me and expressed his bewilderment [1 The remark to be quoted was made by F. Werner when he was a student in Princeton.] with the fact that we make a rather narrow selection when choosing the data on which we test our theories. "How do we know that, if we made a theory which focuses its attention on phenomena we disregard and disregards some of the phenomena now commanding our attention, that we could not build another theory which has little in common with the present one but which, nevertheless, explains just as many phenomena as the present theory?" It has to be admitted that we have no definite evidence that there is no such theory.

The preceding two stories illustrate the two main points which are the subjects of the present discourse. The first point is that mathematical concepts turn up in entirely unexpected connections. Moreover, they often permit an unexpectedly close and accurate description of the phenomena in these connections. Secondly, just because of this circumstance, and because we do not understand the reasons of their usefulness, we cannot know whether a theory formulated in terms of mathematical concepts is uniquely appropriate. We are in a position similar to that of a man who was provided with a bunch of keys and who, having to open several doors in succession, always hit on the right key on the first or second trial. He became skeptical concerning the uniqueness of the coordination between keys and doors.

The hierarchy of ontology and the foundation of mathematics could look like this:

Nothing
Intention
Experience A - Prime memory
Change
Experience A(B): Experience B nested within Experience A
     or with the memory  of A (this implies an experience of identity or oneness nested within the experience of difference)
Dimensions: Similarity / Difference spectrum emerges
    Identity: the "=" sign or the "1"
    Difference: 1 compared to 0
Pattern discovery: digitizing the analog - arbitrarily assigning a similarity threshold to consider Experience N identical to or not identical to Experience N(B)
       the assignation of the similarity threshold is based on intention, purpose, and choice
Pattern overlay: applying the identical arbitrary similarity threshold to find more identities in nested Experience N(C), N(D) N(E), ... etc.
Metaphor: a thread of structure connecting similar differences and different similars
Symbol: a condensed form for a metaphor
The Platonic realm (or as I like to call it, the 5th dimension): the entire symbolic metaphorical structure of all of reality - a map of all pattern overlays 
The Physical realm: an instantiation of the Platonic realm - a symbolic representation of experience - present experience cast into form
Mathematics: a symbolic representation of the Platonic realm.
Physics: Math applied to the physical realm.

Ah I missed this - apologies! But perhaps the Universe was merely keeping it fallow for the moment, because I just read something that seems to potentially tie in with this if I read your ideas correctly.  

Though she removes the Platonic Realm, or at the least shifts it over in the order you have...

The Descent of Math

Sara Imari Walker

A perplexing problem in understanding physical reality is why the universe seems comprehensible and correspondingly why there should exist physical systems capable of comprehending it. In this essay I explore the possibility that rather than being an odd coincidence arising due to our strange position as passive (and even more strangely, conscious) observers in the cosmos, these two problems might be intimately related and potentially explainable in terms of fundamental physics – that is, if we are willing to make the concession that information can physically influence the world. My premise begins by distinguishing physically possible states of the world, defined as everything permissible by the laws of physics, from physically accessible states of the world, defined as those that are achievable from a given initial state. For universes where the laws of physics are such that the number of accessible states is less than all that is possible, I argue that the most probable states among all possible states are those that include physical systems which contain information encodings – such as mathematics, language and art – because these are the most highly connected states in the state space of everything that is possible. Such physical systems include life and - of particular interest for the discussion of the place of math in physical reality - humans. Within this framework, the descent of math is a natural outcome of the evolution of the universe, which will tend toward states that are increasingly connected to other possible states of the universe, a process greatly facilitated if some physical systems know the rules of the game. I therefore conclude that our ability to use mathematics to describe, and more importantly manipulate, the natural world may not be an anomaly or trick, but instead a natural part of the structure of reality.

It also reminds me of this Aeon Essay, suggesting the Mathematical Forms are in *this* world rather than beyond:

The Mathematical World

James Franklin

Still, despite its clean lines and long history, Platonism cannot be right either. Since the time of Plato himself, nominalists have been urging very convincing objections. Here’s one: if abstracta float somewhere outside our own universe of space and time, it’s hard to imagine how can we see them or have any other perceptual contact with them. So how do we know they’re there? Some contemporary Platonists claim that we infer them, much as we infer the existence of atoms to explain the results of chemistry experiments. But that seems not to be how we know about numbers. Five-year-olds learning to count don’t perform sophisticated inferences about abstractions; their contact with the numerical aspect of reality is somehow more perceptual and direct. Even animals can count, up to a point.

In any case, the problem with Platonism is not so much about knowledge as about its view of mathematical entities. Surely when we measure, or calculate, or model the weather mathematically, we are dealing with mathematical properties of real things in this world, such as their quantities. Such properties are not abstracta: like colours, they have causal powers that result in our seeing them. The visual system easily detects such properties as the ratio of your height to mine (if we stand next to each other). There is no room for abstracta in other worlds to enter the story, even if they did exist.

Nominalists and Platonists have fought each other to a standstill, each convincingly revealing the fatal flaws in their opponents’ views, each unable to establish their own position. Let’s start again.

More grist for the mill - I've spent years thinking about Platonism so I have many links on the subject :

Kurt Gödel and the romance of logic

But why should anyone who is not a logician be concerned? Other minds that are both especially mathematical and especially creative have suggested some sweeping—and self-evidently important—applications. The theoretical physicist and mathematician Roger Penrose, for example, has argued that Gödel’s theorem shows that “Strong AI” is false: our minds cannot be computers, and that by extension the intelligence of computers will never fully replicate them. But there is another, more general, way of grasping the importance—and that starts with the broader question of Platonism. Not even Gödel thought that his incompleteness result proved Platonism to be true, but he did think it lent it some support. How? By seeming to undermine one important element of many anti-Platonist perspectives—the view that, in mathematics, truth and provability are the same thing.

This idea that the two are equivalent is associated with the image of the mathematician as an inventor rather than a discoverer. If—as apparently instead established by the “Incompleteness Theorem”—it is possible, in any given system, for there to be an arithmetical statement that is true in that system but not -provable in that system, then one might infer that truth and provability cannot be the same thing. Until and unless the anti-Platonist can find something wrong with Gödel’s proof, he or she seems to be left with the task of finding some way of interpreting it that does not involve accepting the idea that the truth of an arithmetical statement is independent from its proof, a notion that seems awfully close to the Platonic conception of arithmetical truths being (in some sense) “out there.”

Whether this challenge can be met is still—nearly 80 years after the Königsberg Conference—a matter of controversy. I think it is fair to say that many philosophers of mathematics simply do not know what to say in the face of it. Both Russell and Wittgenstein, two of my biographical subjects who were among the cleverest people of their generation, failed to understand it. So, if you find Gödel’s work and its consequences hard to make sense of, don’t worry: you are in good company. Nevertheless, it seems likely that the time will come when this emaciated genius is given his full place in the cultural spotlight alongside his close friend Einstein and his brief acquaintance Russell. Both provided the inspiration and the context for his peculiarly impenetrable, philosophically baffling, and arguably fundamentally important contribution to intellectual life.

=-=-=

Nova PBS -> Math: Discovered, Invented, or Both?

Mario Livio explores math’s uncanny ability to describe, explain, and predict phenomena in the physical world.

Note that if you scroll past the video there's a transcript that can be read/referenced.