Mathematical universe or mathematical minds?
The question seems a correspondent of the most popular question “Was mathematics invented or discovered?” and relates to the nature of mathematics as well as to the philosophical problem of applicability of mathematics. However, there are anthropocentric and evolutionary features that the philosophical investigations on this topic have not focused on much:
https://medium.com/@cb_67963/human-mathematics-and-gods-mathematics-682ac8e7bba
https://medium.com/@cb_67963/human-mathematics-and-gods-mathematics-682ac8e7bba
Comments (72)
I don't quite understand what you're asking. Care to explain a bit better?
Truth be told, math seems to bear cultural signatures e.g. the Greeks were anti-infinity and couldn't fathom how nothing could be something (zero). The Indians were more receptive with respect to both these concepts/ideas. The Chinese were using negative numbers. :chin:
I agree with the linked article that perceptual processes are the place to look for the basis of mathematical reasoning, but rather than ‘innate’ I’d suggest instead they perceptual processes are constructive, forming mathematical concepts as metaphors arising from embodied perceptual interactions with the world.
Interesting article. Thanks. I don't think it will convince anyone one way or another on the issue of the nature of mathematics. I come down on the side of math being a human invention. There are studies that indicate that numerical ability may be present very early in a baby's development.
1.
The massive amount of "pure" mathematical knowledge produced in the last three hundred years suggests this statement is unsupportable.
2.
It would appear from this statement that "Nicholas Bourbaki" is an actual individual, when in fact it's a name a group of math people created when pooling their resources and producing a number of respected textbooks.
Nevertheless, an entertaining read. Thanks.
Quoting 180 Proof
Pure math stems from explorations in applications frequently, though not exclusively. The stuff I have done for years was pure up front and has remained so. It's a combination of discovery and invention.
From Wiki:
logic is a processing structure encoded genetically into our mind which comes from the way reality moves
math is a reduced abstraction of reality, based on logic(the way reality moves)
Glad you said that - I had the same thought.
The idea of naturalizing mathematics is not new. It is how the thesis that mathematics and mathematical truth are discovered (as opposed to constructed or pulled from an ideal Platonic realm) is often cached out.
Though the research in "perceptual mathematics" cited in the article is recent, the general finding that there are innate proto-mathematical capacities should not come as a surprise. This doesn't resolve the question of whether mathematics is invented or discovered, but perhaps the question should be dissolved as a false dilemma. We might gravitate towards certain mathematical structures due to innate predispositions. We also invent mathematics to deal with empirical problems. We also invent mathematics with no practical goal in mind and then, having a ready-made tool at our disposal, opportunistically find a use for it. Nowadays we also invent a load of completely useless mathematics, of which perhaps a small fraction will ever find an application, and the rest will gather dust in mathematical journals and specialist books. Then again, pure mathematicians share the same cognitive apparatus with the rest of humans, they develop in largely the same environment, and their work is influenced by past mathematical culture.
So, what to make of this tangle? That it's not either-or - it's both and then some.
I agree. Many years ago I was on a USAF Office of Scientific Research grant. At the time I was grateful for the financial support, but I wondered why they would fund the sort of things that interested me.
I mentioned 135 math research papers a day received at ArXiv.org. Today it was 269.
What motivates all those math people? Tenure/promotion considerations. Prestige within a community. Delight in the exploratory aspects of a subject with few constraints arising from the physical world - free rein for one's imagination.
From my vantage point as a very senior citizen, the first thing I note is the huge number of people pursuing activities compared with 60 years ago. I haven't a clue as to numbers of mathematicians then and now. But at that time the outdoor sport I became involved with had perhaps a couple of thousand fairly serious devotees here in the USA. Now there are well over six million. World-wide there may be ten million or more. It staggers the mind.
Yeah, I think imagination, curiosity and play are underestimated in these reductionist accounts of mathematics, even though they are as much a feature of our psyche as anything else.
Quoting jgill
Ha! You think there is a connection? :) From my own experience, I've known a few physicist and astronomer climbers, but can't recall any mathematicians off the top of my head.
Not in the way you put it. Philosophy asks a question in a different sense because reality, to philosophy, can be inquired upon in a different sense. However philosophy draws empirical examples and evidence from science.
Yes, I do think there is a connection. I knew and climbed with ten or so fellow math guys from different locales over the years. For me it was problem solving and exploration. Short rock climbs are frequently referred to as "problems". A great combination of intellect and athletics.
That's nonsense. It's the same as saying God is genetically programmed in our genes because that's how reality is. Math isn't programmed by our genes. It's just a way of looking to nature.
point is logic is abstraction related to empiricism
Yes. And?
the brain is not a crystal ball. it has a genetic structure
there is no free will, only biological determinism
what you dont have the genetics to process you will never understand
Only the DNA in my neuron nuclei have. The processes on my neuron network can "resonate" with processes in the real world. But they just as well go contrary. They are not programmed by my DNA. It just happens or not.
before you were born you were nothing but a strand of dna
there was no brain
the dna created the brain as it wanted it to be. then later the brain is born and then it is completely bound by its genetic reaction to the environment. neither of which it chose.
its reactions shape it which then reacts again and shapes it more. over and over until you die
causation
I was nothing but the stuff around it. I just used my genes to develop. They only gave me proteins. They were not involved in any programming. My brain can act like a computer but it isn't one. The physical world can resonate in the brain. This resonance can be structured by math. But only for certain well defined experiments to fit the math. Most of the physical world can't even be approximated by math. There is no math formula corresponding to a piece of music. Any attempt to fit all phenomena in math is doomed.
a piece of music is nothing but a pattern. you just call it music since it alleviates your boredom and raises your dopamine
That's not why I listen to music. That's how you see it. A sad, almost terrifying fact. The soundwave pattern of music cannot be thrown in a mathematical formula. Only short pulses of music can. So what use has math? Not to mention the feeling you get when listening. Just a pattern on the neural network resonating with the music waves. But it feels great! How you explain that?
masterbation feels great
im one with god
But the pattern of sound coming from them can't be caught into a superposition of sines. Unless the music consists out of sine waves in the first place. Can an arbitrary piece of music be Fourier transformed? Only short pieces, not? Short pieces compared to the wavelength. Music pulses.
Now that feeling comes close to a mathematical contemplation.
I don't necessarily think thats correct. Music is sound, and sound is illustrated by mathematics, therefore it should follow that you can illustrate music by math, and math by music.
I have read most of the article your link leads to. Among other things it talks about "non-mathematician mathematicians" which can be only taken figuratively, since obviously it is quite a conflicting expression! Anyway, I can't see what is the point the author of the article the position is trying to make besides that we are all mathematicians and we apply math in our everyday life. Well, this sounds like what I say sometimes about philosophy, namely that everyone is a philosopher and has a philosophy on life and various subjects. But I think we have to address Math as a scientific subject, i.e. the discipline and study of numbers, formulas, relational structures, shapes, etc.
For example, are we inventing mathematical formulas or discovering them?
So, instead of non-mathematicians, let's talk about Mathematics and other sciences themselves.
Now, since I am not a scientist myself, first I bring in some data I found from a short research I did. Then I will tell my views as a non-scientist, from a philosophical view (even if I am not an actual philosopher myself :smile:)
The term "Mathematical universe" leads to "Mathematical Universe Hypothesis", according to which "the physical universe is not merely described by mathematics, but is mathematics (specifically, a mathematical structure)" (https://en.wikipedia.org/wiki/Mathematical_universe_hypothesis).
"Mathematical universe" also leads to Max Tegmark, who wrote a book entitled "Our Mathematical Universe".
Tegmark says, "our external physical reality is a mathematical structure. That is, the physical universe is not merely described by mathematics, but is mathematics (specifically, a mathematical structure). (https://en.wikipedia.org/wiki/Mathematical_universe_hypothesis)
Tegmark explores the possibility that "math does not just describe the universe, but makes the universe."
(https://www.scientificamerican.com/article/is-the-universe-made-of-math-excerpt/
(The book is discussed in detail at https://space.mit.edu/home/tegmark/mathematical.html)
***
Now, about my view on the subject:
Maybe we should start with talking about numbers, the basis of Math. It is supposed that the numeral system was discovered by Egyptians, but this is not important in this topic. It only shows that numbers are apparently human-made. So, we have to ask ourselves, is there a numerical system in the physical universe --independent and different from ours-- based on which the universe "works"? For example, there are three trees out there in the field. Does it matter for the universe? Does the universe use this as information to "act" or "behave" in some particular way? Would it matter if there were four trees or one tree or none? We certainly cannot say.
But this is something more or less concrete. Can such a concepts like "zero", "infinite", etc. have a meaning for or application by the universe? What about "calculus", "combinations" and hundreds of other math methods? I mean, not as terminology but what these represent? In other words, if the nature of the universe or a characteristic of it is mathematical, if the universe has its own way of using what we call "mathematics", how could we ever understand it?
So, on a purely logical basis, a "mathematical universe" makes no sense to me. On the other hand, a "mathematical mind" does.
It's a stretch, isn't it? I think Tegmark's ideas somehow hinge upon the notion of "isomorphism" - the physical universe is the "same" as a purely mathematical structure up to an isomorphism. But I haven't really read his works.
Somehow, this makes sense. A lot of mathematical forms have an isomorphic material counterpart.
Thanks for reminding me of this term! It's quite a long time since it has disappeared from my view ... Well, who knows, there may be some analogy between our mathematics and some inherent system in the universe ... If something like that is discovered, it will certainly be a huge scientific revolution. (Anyway, I will certainly not be here to enjoy it! :grin:)
I reckon, the universe being mathematical and all, atheists won't appreciate it if math were invented. Who invented it?
If math were discovered i.e. math is a natural aspect of the universe, the fact that we're in two minds regarding whether it's invented/discovered is, again, bad news for atheists. Looks invented!
Lose lose atheists, lose lose!
A hypothetical study
Question: How much are you willing to contribute to saving 200, 2000, 20000 migratory birds?
Answer: $80, $78, $88 respectively.
Bird numbers increasing by a factor of × 10.
Contributions pledged: No such pattern.
---
Is the universe really mathematical?
Are humans really good at math?
Are we using a nonmathematical tool that we haven't yet found out exists in our toolkit?
:up:
That article is fascinating, but I can't help but to object to this part:
This is a bold claim... that all pure math is eventually applied. Really? I don't think it's that difficult to program computers to both generate systems of axioms and then crank out theorems.
It makes sense to expect practical math to get more funding than unpractical math. Aesthetics plays a role, surely, but perhaps we are tuned by evolution to appreciate the beauty of an efficient and graceful syntax.
I've commented on this before. ArXiv.org receives 150 - 300 math papers a day, most probably pure math that vanishes into the academic aether after a while having served its purpose, tenure, promotion, prestige within specialties, curiosity, etc.
We wouldn’t need an evolutionary explanation if ‘beauty’ ‘efficient’ and ‘graceful’ can be understood as self-grounding concepts. If they can’t, then they must be dumped in favor of what evolutionary process implies: selection of adaptive concatenations of arbitrary causal mechanisms.
:up:
Is there such a thing? I lean toward Saussure's notion of a system of differences without positive elements. Concepts are only sold in sets.
Quoting Joshs
Why arbitrary? Dennett's vision of a evolution as an algorithm makes sense to me. It's true that neutral traits can come along for the ride (so there's some randomness), but surely there is real selection too.
How could that be? What is the relation, the cause, the link between you and "Doru B"? It looks like magic. Spelling as casting a spell.
But that's arse about. We construct language to be about the world. It is odd, then, to be surprised to find that the world can be set out using language.
Mathematics is stringing symbols together in interesting patterns. Some of those patterns are useful.
Do you mean something like models fitting data pretty well?
:up:
Dennett seems to want to have his cake and eat it too. He says we can understand organic and cultural evolution from within a physical, design or intentional stance. One gets the impression the physical stance is more fundamental for him than the other two. Neural networks composed of dumb bits doing dumb causal
things leads to what , within the intentional
stance, we can call purposeful behavior. But purposeful behavior must be considered fundamentally arbitrary if it is merely the product of such randomly acting bits. Concepts like evolution , order and beauty are ‘higher-order’ products of these primary processes, but how are they any more justified than any other concepts associated with the intentional stance? We can talk ‘as if’ there really is an evolution of order but the meaning of such a notion vanishes within the physical stance. What could an algorithm, much less its evolution , possibly mean within the physical stance?
What is arbitrary doing in that sentence?
If something is an algorithmic process it's not random, and hence not arbitrary. If something is physical, it's not based on a personal whim, and so is not arbitrary.
Evolution is not a random process.
As I see it, it is the claims that apply concepts like evolution which are more or less justified in terms of the usual scientific/rational norms. This is what Brandom calls the primacy of the propositional, and he credits Kant for foregrounding it. We don't build claims from concepts. We understand concepts in terms of the role they play in claims (the inferences they license, etc.)
As I understand 'the physical stance,' it's to be expected that such notions vanish, but the stance is self-consciously reductive. It's a lens that's more or less useful and appropriate in this or that context.
In case it's helpful, I'm happy to grant that Dennett does not know the quiet secret of the universe. We find ourselves here in the mess together (the nightmare of history), and we slowly and painfully work toward being less ignorant and confused, largely by thinking about thinking.
An algorithm produces a lawfulness through the recursive repetition of its formal structure. Where does the algorithm’s formal structure come from? Is it an irreducible a priori or is it the product of a non-algorithmic causal process? If the latter, do we say that the non-arbitrary order of the algorithm emerges somehow out of a process that does not have its order?
Apokrisis wrote a fair bit in previous threads about the gap between the dependence of biological and psychological phenomena on semiotic codes and algorithms vs the absence of the concept of semiosis in physics. One is left with either a kind of dualism in which semiosis appears out of nowhere in living systems or a pan-semiotics inclusive of physics , requiring an updating of meta-theoretical assumptions in physics.
Maybe it's best to talk more concretely. Imagine a chaotic soup of items which are capable of being arranged in self-replicating structures. Perhaps such arrangements are relatively rare, but once they appear they'll tend to say, precisely because they replicate themselves. If such replication is not perfect and includes mutations, it may be that some mutants are more effective self-replicators than others (perhaps most mutations prevent replication.) The essence seems to be that 'progress' is 'saved' or cumulative. We tend to find patterns that are good at hanging around hanging around.
What's wrong with this view? Why couldn't chance invent something that thereafter defies chance as much as it can manage the job ? Self-sustaining, durable patterns are what we'd expect to find even.
This is interesting. I’m on a Joseph Rouse kick, reading his Articulating the World. The book is about thinking about thinking , more specifically thinking about scientific thinking. He critiques authors like McDowell, Brandom and Dennett for not taking advantage of the latest models from biology to ground our scientific/rational norms.
He understands this in terms of a difference between b1and b2 accounts of intentionality:
B1: normative-status accounts of how the performances of a system or group of systems as a whole mostly conform to a systematically construed ideal of rationality
in context, such that the goals with respect to which it would be rational are appropriately taken as authoritative for it
B2: normative-status accounts of how a system’s actual engagement with its surroundings is articulated in a way that renders it accountable to something beyond its
own actual performances or those of its larger community of intentional system
Rouse’s b2 account treats scientific/rational norms as the manifestations of biological niche building rather than as a realm that stands outside of the empirical phenomena that it makes claims about. Claims are performances within a niche of intersubjective practices , just as the normative functioning of organisms defines its environment, changes its environment and is then shaped reciprocally by that changed environment.
This approach rids us the the gap between normative claims ( manifest image) and the empirical world it addresses (scientific image).
“Orthodox and liberal naturalists identify “the scientific image” as a position within the space of reasons, a body of claims that have been justified and accepted scientifically, or as I earlier quoted Price, “the sum of all we take to be the case.” Scientific understanding in practice is instead an ongoing reconfiguration of the space of reasons, of what can count as intelligible and significant projects, defensible positions, reasons for or against them, and possible ways of extending or revising them. Science offers not a single “image” of the world, but a conceptual space of research opportunities and intelligible disagreements.” (Beyond Realism and Anti-Realism At Last)
This sounds right, but I don't really see the conflict. We can choose to use 'scientific image' to refer to a set of relatively settled beliefs while insisting that the process for generating such beliefs is far messier, including at the least lots of unsettled candidate beliefs.
I think that's Sellars' explicit goal. If we imagine a species evolving a second-order tradition of norms for establishing beliefs (a way of talking and acting the world), then we are half way there? Or more?
I think the key term here is ‘imagine’. Without some implicit normative overview transcendent to the phenomena being described we can’t get from ‘chaotic soup’ to ‘self-replicating pattern’. What is it in the phenomena that differentiates chaotic interaction from replicative self-identity or self-similarity? If we reduce material events to processes that have no meaning apart from locally assigned properties, then pattern and self-similarity are concepts that we must bring to events
from somewhere else. For realists this elsewhere is a metaphysical presupposition. For Rouse, normativity is a property of systems of material nature rather than a mind split off from nature.
I think I know what you mean, and I agree. It's only after evolution has happened that its story can be told. But is this more problematic than theories of the cooing of the earth, of a time before such theories were possible ? Must the concept of a supernova exist before actual supernovas occur ? I can understand arguing either side, but I'd butter this bread on the third side, on the brown loop of the crust.
I think that's how Sellars sees it too, and I think I agree. But it's convenient to talk about this or that piece or aspect of nature. 'Rational' humans are just acting in certain ways, caught up in a fragile but potent tradition, more complex perhaps than the culture appearing here and there among other animals, but no more magical or unnatural.
For Rouse, what allows our species to do science is language, but language is homologous with the forms of responsive situational intentionality other animals possess. Other species enact intentional norms but lack our linguistic capability for self-reflection. Science is not centrally about epistemological belief but performances that continually define what is at stake and at issue within a set of partially shared scientific practices. Our performances enact normative pattens just as other living self-organizing systems assimilate their environment to their own normative functioning in relation to their constructed world , while accommodating those norms to the changing circumstances that their own behavior produces in their niche. In other words, science isnt representational, it is enactive.
“ Niche construction theory thus situates conceptual normativity centrally within the evolutionary process in scientifically intelligible ways. It can account for not only the continuities between our conceptual capacities and the flexible, instrumentally rational responsiveness of many other organisms to their developmental, physiological, and selective environments but also for the crucial discontinuities between them. We are adaptively and reconstructively responsive to a very different environment, which has coevolved with our conceptual capacities. The key transformation was the development of partially autonomous performative and recognitive repertoires through the ability to track and assess them in two dimensions. We are responsive to a dual significance of various performances and circumstances, both for appropriateness within their proximate domains and for their broader significance for our lives and ways of life.”
This seems to get things right enough. Far more than beavers, we create the world we study as our study gives us more and more power to shape that world. We also create and edit our own norms, but only in the light of the norms we have so far. It's like Neurath's boat. We can never question all of them at once, but only some of them in terms of a majority of others necessarily left unquestioned.
Turning this on its head, there are those who argue instead that the complex numbers are more fundamental than the reals because they embed the seed of commutativity that Nature needs to physically exist.
Metaphysics requires symmetry breaking or asymmetry to get a Cosmos going. And complex numbers make commutative order matter in a way that is "physically realistic". The reals are just too simple in that they lose this grit which eventually forms the pearl.
So this is a bit of a parable that shows it isn't an either/or situation. Maths and reality are in a dialogue as far as our epistemic endeavours go. In this case, the mathematical intuition was that the reals had to be fundamental, so "complex number magic" became "a surprise". But if we had instead started from more metaphysical considerations – the needs of a world formed by symmetry breaking - then complex numbers might have come first, and the reals be considered the "less fundamental" afterthought.
Quoting Joshs
I can go along with just about any criticism of Dennett, but this could be harsh to physicalism which after all now founds itself on deeply holistic principles like general covariance and least action. There is a finality, a Darwinism, in effect that selects for the cosmic structure that best "hangs together".
Again, this is where we can look for the robust connection between "maths as invented" vs "maths as real". We are merely creatures making models. But the structures that are useful to describe are the ones by which the Cosmos must inevitably structure itself. So we do a good or bad job in that regard.
Quoting Joshs
I would use the term pansemiosis as a synonym for dissipative structure theory and hierarchy theory. That is, all these are attempts by the biologically-inclined to root their biology in a physics that is triadically complex rather than monistically simple.
So semiosis is dependent on the extra thing of an encoding mechanism - genes, neurons, words, numbers. That is something completely new to Nature - and yet also already "existent" in Nature as that which is antithetical. Symbols have their unbounded power over physics because they essentially zero the cost of regulating that physics. Semiosis transcends that which it seeks to control by placing itself outside the material cost of doing business - or at least by making the cost so tiny in comparison to the returns that it drops out of the equation.
Once you have a mechanism for constructing proteins, you can make any protein at all. Useful ones, useless ones. It's all the same. And thus the proteins you make become a meaningful choice.
Nature “in the raw” lacks this self-transcendence. It just self-organises. And we can call that pansemiosis because it is the step that paves the way for semiosis proper as its "other".
Quoting Joshs
Yep. Semiosis does the extra thing of imposing its imagined regulative possibilities on a world that has the clear possibility of being regulated.
But this gets confusing when we both need to model the world "as it actually is" so that we can then likewise construct our widest range of possible worlds to impose upon it.
Which is the science and maths suppose to track? Well, it sort of does both if we can disentangle the fundamental view from the applied view.
And yet by the same token, it would serve no point to actually sever the connection between our manifest and empirical worlds as that is the pragmatic connection being nurtured.
So the game is to divide, and then to connect. Semiosis is about constructing the reality we wish to live. That starts down at the genetic level for life. Termites shape their worlds into the world that best befits termites. The result is neatly spaced mounds with great air-conditioning, etc. And humans take that to an anthropic extreme with the world they build for themselves.
So epistemology requires a clean break into the subjective and the objective as the step towards its next level of reality construction. Give humans a chance and they would anthropomorphise not just a single planet but the entirety of the Cosmos.
It can't happen. But if it did, it would be Nature playing out the logic of pansemiosis.
Quoting Joshs
Actually this was the phase that got me perusing this thread. :up:
It is essentially what I am saying. Maths is both the free invention of our minds and the inescapable organisation of any Cosmos. It takes this kind of clean break - this dichotomy that defines two complementary limits – to ground the actual business of semiosis, which is to continue the self-organising evolution of the natural world.
So we need a theory of the world (as it really is) and a theory of the self (as it ideally would be). From the interaction of the two, we get whatever we get.
And to get this clean division of theory, we need the meta-theory that can see this as a pragmatic co-production. A theory of the self especially needs a material grounding - as the current sad state of world shows. And our theory of the world is likewise rather lacking in its material holism, its reliance on dissipative order, etc. The science and maths we favour is on the reductionist side. Short-term in its horizons, simplistic in its interactions.
In terms of the OP, we have a functioning balance of invented~discovered that was good enough to deliver the industrial age based on a fossil fuel "free lunch". That created a stage in the Hegelian advance of history. That manifested a certain concrete reality.
What comes next is its own interesting question. But the semiotic view says that to say anything intelligent, we have to focus on the fact that this is about a self~world modelling relation.
We built ourselves up to our current position on a hierarchy of genes, neurons, words and numbers. Words delivered humans as social selves. Numbers delivered humans as technological selves. Does further progress require a new level of semiotic mechanism - one still more abstract or advanced – as words and numbers seem to trap us in the kinds of self~world structures they are able to create.
Perhaps you are speaking of the canonical commutation rule in QM? Obtained by employing the imaginary number i. Otherwise I see no particular advantage in, say, multiplication or addition over the reals. Of far more interest is Euler's formula and its relation to wave forms.
Alack, these be feelings I'll probably never experience!
It would help to familiarise yourself with the current metaphysical debate over Penrose’s argument. Here is one excellently clear paper….
http://cejsh.icm.edu.pl/cejsh/element/bwmeta1.element.ojs-issn-2451-0602-year-2018-issue-65-article-439/c/439-462.pdf
The article then goes on to cover the gory mathematical details.
What is key is that the complex numbers build in something essential to physical existence itself - something so basic that all more complex structure pops out of it “for free”, whereas getting the same outcomes from real numbers is arduous, as they lack the right dimensionality to describe the dimensionality that the Universe actually has.
As a cosmic Darwinist, this tells me that we live in the Universe with the particular dimensionality that is the simplest way to produce richness, and that is why physics finds that complex number magic is matchingly the root description of Nature.
The reals are too simple to generate a complex world. But the complex numbers have the minimal complexity that can then be the basis for generating all the complexity that ensues.
The hypothesis I am pursuing is that it all ties together in rather obvious fashion with Noether’s theorem and Newton’s twin conservation principles - the dimension-defining dichotomies of rotation and translation.
Dimensionality is a system defined by the local and global. And such a system can be broken towards its two dialectical extremes - the spin that defines localisation, the straight line or flat geometry that defines the globality of unbounded translations.
If your number system has to have units that speak to both unit 1 rotation and unit 1 translation as their scale-anchoring identity operation, then what else would you expect your number system to look like as the simplest possible representation of such a unity of opposites?
So this is why particle physics is cashed out in stories of spin symmetry, and why Penrose pushes so hard with with his hylomorphic/conformal view of spacetime geometry.
(We were speaking of fixed points before. I introduced the idea of accelerating the convergence of limit-periodic CFs through the use of attracting fixed points of LFTs back some years ago.)
So what are your thoughts here when one direction looks to track the "deep maths" of Nature and the other choice may be just unphysical pattern spinning? What do we learn if this is the case?
But am I right that you argue the complex plane has lessons in terms of the physics of chaos - patterns of convergence~divergence?
I couldn't get much out of your 1970s paper, but I was thinking about how your fixed point paper illustrates my point about the dichotomy of rotation~translation in that the complex plane seemed marked by patterns of curl. Or convergence and divergence over all scales, as you would expect in a chaotic system.
In a maximally turbulent system, you have the fixed points of vortexes appearing over all available physical scales. And the the deep naturalness of this scalefree or powerlaw behaviour was rather the point of the Franks paper I cited previously.
So - in reference to the OP - it seems hard not to be interested in the maths that has some connection to reality, even if the ideology of maths is that is perfectly free to chase pure pattern for its own sake.
And coming from the other direction, Nature itself has no choice but to be structured and fall into self-constraining patterns. Mathematical regularity of some kind must emerge.
But I was reading this yesterday that warns the field of maths might indeed be shortchanging itself with some of the metaphysical choices it makes ... yet equally, one could argue that the power of maths lies in the fact it puts itself outside the reality it means to describe by making its "non-intuitionist" choices ... like the ones about infinities and empty sets that outrage the more metaphysically inclined.
So there's a thought. Zoom in on your complex plane with its pattern of curl, and do you start to lose any sense of whether some infinitesimal part is diverging or converging? Is even a maximally curved part of the map now definitely one or the other. So same as the real number continuum. Zoom in and are your cuts a point on one side or the other? Bring in the observer (or observational scale) – as physics must - and the intuitionist position has a lot going for it.
Seeing as I'm throwing out references, here is another that might interest you - a short, then longer version, of Peirce on the continuum debate. One can ask again whether maths made the right pragmatic choice even if Peirce is the metaphysically correct choice? And so everyone is right and wins a prize. :grin:
https://cesfia.org.pe/villena/zalamea_peirce_translation.pdf
https://uberty.org/wp-content/uploads/2015/07/Zalamea-Peirces-Continuum.pdf
The physics and the mathematics go hand in hand, the problems of one are not the cause of the problems of the other, but they do amplify each other. Einsteinian relativity is what expulses time from physics. And since physicists choose to employ this principle, they also choose the mathematical axioms which facilitate the application of it. So as soon as the principle which expulsed time from physics was established, then the axioms which expulsed time from mathematics were applicable, and acceptable
Relativity united time and space in a way that made more general sense. And we now wait for quantum physics to catch up with the rest of the class.
It really does not make "general sense". It's a counterintuitive principle which facilitates calculations when using electromagnetism as a measuring tool.
Relativity is extremely counterintuitive because it describes time as passing at different 'rates', or 'speed', depending on one's physical circumstances. The intuitive way is to conceive of us human beings as being completely incapable of altering the passage of time. We experience time as passing in a way which is completely determined, or fixed by the physical universe, so that we are utterly helpless to slow down the arrival of the future, or alter the past. Relativity, if it is true, gives us the capacity to alter the flow of time through accelerations, and this is very counterintuitive.
Motivations vary in the mathematical community. What is common, however, is the drive to explore, sometimes with regard to the mysteries of nature, sometimes within the discipline itself. I'm a spinner.
Quoting apokrisis
I don't know about the physics of chaos (other than my experiences as a meteorologist ages ago), but the patterns that appear as examples of weak emergence are fascinating.
Since computations are done on computers with computable numbers and functions, isn't that already the case? The notion that a non-intuitionistic approach damages the idea of time seems ridiculous. A purely philosophical tragedy.
I've never known a fellow mathematician who claimed to be an intuitionist. The fact that all functions from [0,1] to R are continuous from that perspective is quite unappealing to someone who came up in classical analysis. (Yes, it depends on the definition of "function"). Equating the flow of time with adding more digits to a number seems a bit absurd, at least for me.
Quoting apokrisis
The complex plane itself doesn't have a pattern of curl. It's a vector field based on a complex function that does the job. I have zoomed in up to 10,000X to display fascinating objects purely from curiosity. In the vicinity of an attracting fixed point, no matter the magnification, one sees convergence. It's just a matter of how one writes the computer program.
Quoting apokrisis
I admire Peirce for his thoughts on nonstandard analysis. I once toyed with the idea of teaching a real analysis course from that perspective, but gave it up when a friend at a larger university did just that, with poor results. I find it a bit amusing that when one looks at the standard graphical depiction of a mathematical Category, one sees Peirce's Triangle.
Some thinkers assume that Nature is Mathematical (abstract values, sans meaning) while others believe that Nature is Mental (logic plus meaning). So, I suspect that the replies to your topic will divide along those lines. A purely mathematical universe just is, and must be taken for granted. That's why many scientists assume that Energy (cause) & Laws (logic) exist eternally, and need no explanation. But some scientists observe that the universe, that began from abstract Cause & Laws, has evolved animated self-aware beings with personal values & meanings. How is that possible?
I'll propose that, logically, the Potential for Awareness & Feelings must have existed, in potential, along with Causation & Organization. Over time, that Potential for both Material and Mental processes has been Actualized into physical Bodies with non-physical Minds. Why do I label Mind as non-physical (or metaphysical)? Because, the reductive sciences of humanity have not yet discovered an Atom of Mind. So, they label Mental processes as "epiphenomena", hence, not nearly as important (for practical purposes) as Physical phenomena.
As a thought experiment, which would you prefer to be : a> a phenomenal Brain in a vat, without awareness & feeling, or b> an epiphenomenal Mind in a vacuum, with sentience & sensations. Your answer will reveal which you believe to be most essential to your being. Of course, a Mind without hands cannot do anything physical. But it can imagine doing anything imaginable. So, which is more important to you : Mechanism or Imagination? :joke:
PS__How about option c> a physical body & brain plus a non-physical mind? Is that the best of all possible worlds?
Epiphenomena :
[i]1. a secondary effect or byproduct that arises from but does not causally influence a process.
2. mental state regarded as a byproduct of brain activity.[/i]
Potential is Primary :
In physics as in Logic, an Action Potential necessarily exists prior to the Action.
Most of the proposed answers I've seen, to the "invented vs discovered" question, seem to conclude that it's a little of both. The "unreasonable effectiveness of mathematics" in science indicates that Nature is in some sense fundamentally mathematical. But the history of math shows that humans using abstract "pure" mathematical principles, eventually find concrete practical applications for many of them.
That's why I have concluded that human Logic (Reason) is basically Geometry with words. Both manipulate abstract relationships (ratios ; relative values) in order to discover wider or narrower practical applications of those ratios. Ratios are inherent in Nature : for example, Energy is essentially a thermodynamic ratio between high & low, hot & cold, here & there. Likewise, Morality is basically an interactive ratio between me & you, us & them, good & evil. So, it seems that Evolution has programmed the human brain with a rudimentary sense of ratio, that we can expand on with education & practice. But, we can also lose that sense, when it is not exercised. Which is why I can no longer add 2 + 2, without the crutch of a non-human calculator.
I just read an article that deals with a similar topic : Human mathematics and God’s mathematics, by Catalin Barboianu, mathematician & philosopher of science. For the purposes of this thread, you can interpret "God's Mathematics" as the logical structure of Nature, and "Human Mathematics" as the human talent for discovering and inventing applications for that knowledge. He seems to believe that "to mathematize is human", so to speak : "Whatever mathematics is (science, method, formal language or logical symbolism), we do mathematics without being mathematicians". Yet, he also offers an alternative theory, that humans project their own subjective values onto Nature.
Speaking of subjective math, he adds that "The history of application of mathematics in the sciences also has a “mysterious” element. Being driven by their natural impulses of inquiring and generalizing, but also following some special criteria of beauty, symmetry and elegance specific to the mathematical creation". Although we can't yet describe the evolutionary mechanics that created the aesthetic sense in humans & animals, it's more taken for granted than viewed as mysterious. Without consciously thinking about it, we can immediately recognize the symmetry & elegance in a beautiful face. And some have even tried to reduce it to geometry. :yum:
"Contemporary pioneering studies in what is called perceptual mathematics . . . . came to shape an interdisciplinary cognitive theory that claims that all mathematics is human, resides in the mind, and is not an external product of the mind. The human mind is endowed with innate primordial perceptions such as spatial (metric, linearity), numerical, and topological (proximity, relational structures), reflected by the common empirical concepts such as distance,motion, change, flow of time, and matter. It is further hypothesized that animals also hold such perceptions.Thus, the concepts of mathematics are not platonic, but are built in the brain from these primordial perceptions, and brain neurophysiology gives rise to the extremely precise and logical language of mathematics
https://medium.com/@cb_67963/human-mathematics-and-gods-mathematics-682ac8e7bba