• jgill
    2.6k
    So, on a purely logical basis, a "mathematical universe" makes no sense to meAlkis Piskas

    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.
  • Raymond
    815


    Somehow, this makes sense. A lot of mathematical forms have an isomorphic material counterpart.
  • Alkis Piskas
    1.4k
    hinge upon the notion of "isomorphism"jgill
    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:)
  • Agent Smith
    8.1k
    Is there no middle ground or a third alternative?

    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!
  • Agent Smith
    8.1k
    Scope neglect

    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?
  • Pie
    1k
    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

    :up:
  • Pie
    1k


    That article is fascinating, but I can't help but to object to this part:

    Well, the history of science has proved thatwhatever complex concepts mathematicians created, they finally came to be applicable in the mathematics of physics or even to directly describe an empirical context. Take for instance the classical example of complex numbers.

    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.
  • jgill
    2.6k
    This is a bold claim... that all pure math is eventually applied. Really?Pie

    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.
  • Joshs
    4k
    perhaps we are tuned by evolution to appreciate the beauty of an efficient and graceful syntaxPie

    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.
  • Pie
    1k
    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.jgill

    :up:
  • Pie
    1k
    We wouldn’t need an evolutionary explanation if ‘beauty’ ‘efficient’ and ‘graceful’ can be understood as self-grounding concepts.Joshs

    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.

    they must be dumped in favor of what evolutionary process implies: selection of adaptive concatenations of arbitrary causal mechanisms.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.
  • Banno
    19.2k
    Imagine someone being astonished that sentences featuring the marks "Doru B" can be used to talk about you...

    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.
  • Pie
    1k
    Somehow, this makes sense. A lot of mathematical forms have an isomorphic material counterpart.Raymond

    Do you mean something like models fitting data pretty well?
  • Pie
    1k
    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.Banno

    :up:
  • Joshs
    4k
    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.Pie

    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?
  • Banno
    19.2k
    But purposeful behavior must be considered fundamentally arbitrary if it is merely the product of such randomly acting bits.Joshs

    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.
  • Pie
    1k
    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.Joshs

    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.
  • Joshs
    4k
    But purposeful behavior must be considered fundamentally arbitrary if it is merely the product of such randomly acting bits.
    — Joshs

    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
    Banno

    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.
  • Pie
    1k
    do we say that the non-arbitrary order of the algorithm emerges somehow out of a process that does not have its order?Joshs

    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.
  • Pie
    1k
    semiosis appears out of nowhere in living systemsJoshs

    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.
  • Joshs
    4k

    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.)

    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
    Pie

    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)
  • Pie
    1k
    “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.”
    [\quote]
    To me it makes sense to speak roughly of the scientific understanding of a place and time, 'the sum of what we [the scientifically educated] take to be the case.'

    Joshs
    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)Joshs

    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.
  • Pie
    1k
    This approach rids us the the gap between normative claims ( manifest image) and the empirical world it addresses (scientific image).Joshs

    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?
  • Joshs
    4k
    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.Pie

    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.
  • Pie
    1k
    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’.Joshs

    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.
  • Pie
    1k
    For Rouse, normativity is a property of systems of material nature rather than a mind split off from nature.Joshs

    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.
  • Joshs
    4k
    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?Pie

    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.”
  • Pie
    1k
    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.Joshs

    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.
  • apokrisis
    6.4k
    Well, the history of science has proved that whatever complex concepts mathematicians created, they finally came to be applicable in the mathematics of physics or even to directly describe an empirical context. Take for instance the classical example of complex numbers.

    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.

    We can talk ‘as if’ there really is an evolution of order but the meaning of such a notion vanishes within the physical stance.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.

    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.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".

    This approach rids us the the gap between normative claims ( manifest image) and the empirical world it addresses (scientific image).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.

    For Rouse, normativity is a property of systems of material nature rather than a mind split off from nature.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.
  • jgill
    2.6k
    And complex numbers make commutative order matter in a way that is "physically realistic"apokrisis

    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.
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