Tegmark's mathematical universe hypothesis

[James R]

Moderator note: this thread is for on-topic discussion of Max Tegmark's "Mathematical Universe" hypothesis.

Some of the posts here can also be found in the following thread, which is more for discussion of a certain member's obsession with Tegmark's hypothesis:

We need more discussion of Tegmark's mathematical universe hypothesis
---

Write4U:

In one of your many threads about Tegmark and his MUP (mathematical universe hypothesis), I linked the following articles:

https://www.math.columbia.edu/~woit/wordpress/?p=6551&cpage=2

https://www.science20.com/rationally_speaking/mathematical_universe_i_ain’t_convinced-127841

There are substantive objections to Tegmark's hypothesis here. Bear in mind, by the way, that the fact that I previously mentioned these makes a lie of your claim that I only ever make ad hominem attacks on you and never provide specific scientific objections to what you post.

How do you respond to the objections raised? You ignored these, last time.

For the historical context of our previous discussion about this, see the following thread. The link is to where I first pointed you to these review articles, back in December 2020:

https://www.sciforums.com/threads/t...ematical-construct.163837/page-2#post-3657387

 

[Halc]

I am pretty much reacting to the topic title and not so much with the debate with Write4U

On the mathematical universe thing, I think you'll probably find that many people on this forum will accept that the universe shows signs of "mathematical order". If it did not, we'd have a lot of trouble explaining why mathematical models of the universe work so darn well.

The MUH goes further than that since it posits that mathematics is more proscriptive, not just descriptive, the territory and not just the map. I am sort of a proponent of the idea, but not particularly of how Tegmark spins it.

I don't know why you add the word "abstract". All mathematics is abstract, isn't it?

Not with the MUH, no. There is no requirement of anything abstracting the mathematics necessary for the universe to be, which would indeed make it a religion since the abstractor would serve the role of the god.

 

[James R]

Halc:

The MUH goes further than that since it posits that mathematics is more proscriptive, not just descriptive, the territory and not just the map. I am sort of a proponent of the idea, but not particularly of how Tegmark spins it.

There's been a lot of previous discussion about Tegmark's MUH on this forum, although Write4U appears to have forgotten most of it.

I am well aware that Tegmark's main claim in his MUH is not that the universe is mathematical, but that the universe is mathematics. That is, he claims that, somehow, "mathematical structures" are what the physical universe is made of.

Personally, I think this is a basic category error - confusing the map with the territory. Your post indicates that you recognise this, and yet you also call yourself a "sort of proponent of the idea"? What do you mean by that? Do you believe that universe is made of maths?

As for Write4U, he started off in lockstep with Tegmark on that, but appears to have backed off from that position over the last two years. Of course, by now he might have forgotten why he backed away from it; it wouldn't surprise me at all if he now wants to reassert that the universe really is just mathematics, again.

Not with the MUH, no.

That's the major problem with the MUH, right there. The question Tegmark needs to answer is how an abstract set of principles or theorems possibly create any sort of physical object. Or, more accurately, how could the substance of any physical object possibly be made of something that is fundamentally a set of ideas or concepts?

There is no requirement of anything abstracting the mathematics necessary for the universe to be, which would indeed make it a religion since the abstractor would serve the role of the god.

I think that a lot of people who wouldn't describe themselves as religious still like to think that there must be some "ultimate foundation" for the universe. Some of those people want to shoe in some sort of Creator being. When they do that, they almost invariably import ideas from mainstream religions, one way or another. The fact that they are doing that, subconsciously or otherwise, becomes more explicit if they go on to things like "God is the ultimate foundation of the universe".

If there is, in fact, an ultimate foundation for the universe, "God" would be a terrible name for it, especially if it's something like a multiversal law of physics, or - as Tegmark would have it - "mathematical structures". Using the term "God" immediately confuses the issue by applying a label with endless historical connotations to something that the believers say has no such connotations (although, it's often difficult to be really sure if they are being honest about what they actually believe).

I would describe Tegmark's view as more of a pseudoscientific one than a religious one. It is notable that the beliefs of Tegmarkian followers like Write4U are more like a religion, because those followers have themselves a Prophet of the faith, whose proclamations they accept essentially without question, as if they are the Word of God.

 

[Halc]

Personally, I think this is a basic category error - confusing the map with the territory

Your suggestion of it confusing the map with the territory presumes a different model. Take the hypothesis on its own assumptions without adding your own.

Your post indicates that you recognise this

I say that the MUH suggests that the map is the territory. It does not commit the confusion you indicate.

What do you mean by that? Do you believe that universe is made of maths?

Belief is a strong word. If there's anything I believe, it is that I am far too ill informed to stand much of a chance of actually guessing about the actual nature of things, and that is assuming that there is an answer, however unknowable.

Yes, the hypothesis makes sense to me, but also has problems. I've not seen most of the other threads so I don't know if they're brought up. The hypothesis says that the universe is a single mathematical structure, which is different than saying the universe is made of maths, which make it sound like it is composed of a bunch of things, each of which is maths. Tegmark goes further and asserts the reality of such structures (all of them, not just ours). He is a strong proponent of MWI which makes a similar realist claim. I have issues with realism and find it the source of far more problems than it seems to solve.
Tegmark is also a dualist of sorts, even if he might not admit it, but his quantum suicide proposal relies on an identity independent of any physical state. I think his book would have been taken more seriously had he left that part out.

As for Write4U, he started off in lockstep with Tegmark on that, but appears to have backed off from that position over the last two years.

Well I've also been all over the map, abandoning positions that I felt have failed to hold up to scrutiny. It's fine to change opinions if the change is for the right reasons. But no, I cannot seem to glean what Write4U actually believes.

The question Tegmark needs to answer is how an abstract set of principles or theorems possibly create any sort of physical object.

I don't think Tegmark posits that the universe is an abstraction. An abstraction (to me) is a product of something doing the abstracting, like a simulation or mind or something. A simulation is not a mathematical object, it is simply a machine. Such a proposal breaks down into idealism of sorts. I don't think mind is fundamental. Quite the opposite, not that it is absurd, but that if true, nothing can be learned, rendering it empty. Epiphenomenalism is similarly empty, rendering itself false merely by suggesting the truth of it.

Or, more accurately, how could the substance of any physical object possibly be made of something that is fundamentally a set of ideas or concepts?

I think you confuse MUH with idealism, which suggests pretty much that. There is no fundamental thing with ideas or concepts in the MUH.

 

[exchemist]

Write4U:

In one of your many threads about Tegmark and his MUP (mathematical universe hypothesis), I linked the following articles:

https://www.math.columbia.edu/~woit/wordpress/?p=6551&cpage=2

https://www.science20.com/rationally_speaking/mathematical_universe_i_ain’t_convinced-127841

There are substantive objections to Tegmark's hypothesis here. Bear in mind, by the way, that the fact that I previously mentioned these makes a lie of your claim that I only ever make ad hominem attacks on you and never provide specific scientific objections to what you post.

How do you respond to the objections raised? You ignored these, last time.

For the historical context of our previous discussion about this, see the following thread. The link is to where I first pointed you to these review articles, back in December 2020:

https://www.sciforums.com/threads/t...ematical-construct.163837/page-2#post-3657387

I too have quoted Peter Woit's criticism of Shapiro/Tegmark in past discussion about the MUH (as in meh?).

I find Woit and Massimo Pigliucci provide useful critiques of some classes of metaphysical extrapolation.

 

[exchemist]

Your suggestion of it confusing the map with the territory presumes a different model. Take the hypothesis on its own assumptions without adding your own.

I say that the MUH suggests that the map is the territory. It does not commit the confusion you indicate.

Belief is a strong word. If there's anything I believe, it is that I am far too ill informed to stand much of a chance of actually guessing about the actual nature of things, and that is assuming that there is an answer, however unknowable.

Yes, the hypothesis makes sense to me, but also has problems. I've not seen most of the other threads so I don't know if they're brought up. The hypothesis says that the universe is a single mathematical structure, which is different than saying the universe is made of maths, which make it sound like it is composed of a bunch of things, each of which is maths. Tegmark goes further and asserts the reality of such structures (all of them, not just ours). He is a strong proponent of MWI which makes a similar realist claim. I have issues with realism and find it the source of far more problems than it seems to solve.
Tegmark is also a dualist of sorts, even if he might not admit it, but his quantum suicide proposal relies on an identity independent of any physical state. I think his book would have been taken more seriously had he left that part out.

Well I've also been all over the map, abandoning positions that I felt have failed to hold up to scrutiny. It's fine to change opinions if the change is for the right reasons. But no, I cannot seem to glean what Write4U actually believes.

I don't think Tegmark posits that the universe is an abstraction. An abstraction (to me) is a product of something doing the abstracting, like a simulation or mind or something. A simulation is not a mathematical object, it is simply a machine. Such a proposal breaks down into idealism of sorts. I don't think mind is fundamental. Quite the opposite, not that it is absurd, but that if true, nothing can be learned, rendering it empty. Epiphenomenalism is similarly empty, rendering itself false merely by suggesting the truth of it.

I think you confuse MUH with idealism, which suggests pretty much that. There is no fundamental thing with ideas or concepts in the MUH.

One problem I have with the idea of the MUH is that one can only apply mathematics to physical concepts that one has first defined in words. There must, it seems to me, be physical entities, with physical properties, which stand outside the mathematics and to which a mathematical framework can be applied.

 

[Halc]

One problem I have with the idea of the MUH is that one can only apply mathematics to physical concepts that one has first defined in words.

But the MUH is not about concepts, and thus there is no such restriction. This seems to be a straw man objection.

You seem to want to require something more fundamental, more 'physical', from which the mathematical relations can emerge, but that begs a form of physicalism, not the MUH at all.

The objects I've heard that seem to have teeth seem to require a decent backing in probabilities and a good understanding of things like Boltzmann brains. The general argument notices that the vast majority of mathematical objects are junk, random and uninteresting. Any hypothesis that makes it more probable that we are one of those pieces of junk cannot be taken seriously since there can be zero empirical evidence of anything. That's a hard criticism, and one for which I have no easy answer.

As for Tegmark's position, it is one of several proposals of mathematical Platonism, and it is that very Platonism to which I object. My version of MUH is not one of platonism, but more of a relational view, and it is not presented at a level of an assertion, merely something that seems more plausible to me than any alternative I've found.

 

[exchemist]

But the MUH is not about concepts, and thus there is no such restriction. This seems to be a straw man objection.

You seem to want to require something more fundamental, more 'physical', from which the mathematical relations can emerge, but that begs a form of physicalism, not the MUH at all.

The objects I've heard that seem to have teeth seem to require a decent backing in probabilities and a good understanding of things like Boltzmann brains. The general argument notices that the vast majority of mathematical objects are junk, random and uninteresting. Any hypothesis that makes it more probable that we are one of those pieces of junk cannot be taken seriously since there can be zero empirical evidence of anything. That's a hard criticism, and one for which I have no easy answer.

As for Tegmark's position, it is one of several proposals of mathematical Platonism, and it is that very Platonism to which I object. My version of MUH is not one of platonism, but more of a relational view, and it is not presented at a level of an assertion, merely something that seems more plausible to me than any alternative I've found.

Yes, that's fair comment. I do think mathematics is abstract and thus, at its root, not physical. I cannot see how mathematics alone can generate a world of physical experience.
Mathematics, surely, is about relationships between quantities. Those quantities we associate with physical concepts (physical entities and their attributes) need to be defined before we can know what mathematical relations apply to them.

 

[exchemist]

Do you actually understand what Tegmark is asserting with his MUH, and what he means by a "mathematical universe"? Would you like to summarise your understanding, at least, so that we can at least be sure that when you throw his name around we can be confident that you do so correctly? Because unfortunately we lack that confidence at that moment.

My problem with the idea is rather along the same lines as Pigliucci's difficulty with it: https://archive.ieet.org/articles/pigliucci20131212.html

QUOTE

One obvious problem is posed by what it would mean for the world to be “made of” mathematical structures. The notion of mathematical structure is well developed, so that’s not the issue. A structure, strictly speaking, is a property or a group of mathematical objects that attach themselves to a given set. For instance, the set of real numbers has a number of structures, including an order (with any given number being either less or more than another number), a metric (measuring the distance between points in the set), an algebraic structure (the operations of addition and multiplication), and so on.

The problem is in what sense, if any, can a mathematical structure, so defined, actually be the fundamental constituent of the physical world, i.e. being the substance of which chairs, electrons, and so on, are made.

Of course, both Julia and I asked Max that very question, and we were both very unconvinced by his answer. When Tegmark said that fundamental particles, like electrons, are, ultimately mathematical in nature, Julia suggested that perhaps what he meant was that their properties are described by mathematical quantities. But Max was adamant, mentioning, for instance, the spin (which in the case of the electron has magnitude 1/2). Now, the spin of a particle, although normally described as its angular momentum, is an exquisitely quantum mechanical property (i.e., with no counterpart in classical mechanics), and it is highly misleading to think of it as anything like the angular momentum of a macroscopic object. Nevertheless, Julia and I insisted, it is a physical property described by a mathematical quantity, the latter is not the same as the former.

Could it be that theories like MUH are actually based on a category mistake? Obviously, I’m not suggesting that people like Tegmark make the elementary mistake of confusing the normal meaning of words like “objects” and “properties,” or of “physical” and “mathematical.” But perhaps they are making precisely that mistake in a metaphysical sense?

UNQUOTE

(I don't know who Julia is, by the way.)

 

[Write4U]

Do you actually understand what Tegmark is asserting with his MUH, and what he means by a "mathematical universe"? Would you like to summarise your understanding, at least, so that we can at least be sure that when you throw his name around we can be confident that you do so correctly? Because unfortunately we lack that confidence at that moment.

Yes and that is only a result of rumor.

Yes, I watched Tegmark's lectures. I got my knowledge directly from the "horse's mouth". Have you?
Let's start with Tegmark's hypothesis:
Starting from 19:00

Now, I don't like Tegmarks use of the human terms "numbers" .
I believe a more objective approach is the use of the term "values" and "patterns".. That makes a clear distinction between the concept of interactions between human symbolic numbers and naturally occurring generic relational "values" and self-organizing "patterns".

Of course, none of this makes any difference in the concept of the universe as a mathematically guided physically functioning object. I believe that most of the science acknowledges some mathematical properties to universal phenomena.
Tegmark just took it a step further.

 

[Pinball1970]

So three threads? This on mathematics to

At a high level, of course. If you would be so good? And no, reposting that video, or any where someone else explains it, will not suffice in showing that you understand it.

I watched the video and wondered what the hell it was!
Probably because he did not really mention mathematics till 25 minutes into a 45 minute video.
We got a brief summary of where we are in current cosmology including sped up video of a radio telescope for the first 25 minutes.
The last ten minutes was a discussion of the future, what we may make of it.
All good stuff BUT what did he actually say about the universe and mathematics? What was his overall thesis? The Universe IS mathematics....Is that it?
Right, how does that work? What does that mean?
We can always buy his book of course (available at most reputable bookstores...)

 

[Sarkus]

So three threads?

Such is the ability of Write4U to introduce numerous irrelevant strands into a topic that it needs three more threads to untangle them. And even in those he struggles to keep it on topic. Ah, well.

I watched the video and wondered what the hell it was!
Probably because he did not really mention mathematics till 25 minutes into a 45 minute video.
We got a brief summary of where we are in current cosmology including sped up video of a radio telescope for the first 25 minutes.
The last ten minutes was a discussion of the future, what we may make of it.
All good stuff BUT what did he actually say about the universe and mathematics? What was his overall thesis? The Universe IS mathematics....Is that it?
Right, how does that work? What does that mean?

Congratulations on sitting through the video. You're a better man than me. I'll wait for the precis from the person who seems to expect us to do so, so that he can demonstrate that he at least understands what the video is about etc.

 

[exchemist]

So three threads? This on mathematics to

I watched the video and wondered what the hell it was!
Probably because he did not really mention mathematics till 25 minutes into a 45 minute video.
We got a brief summary of where we are in current cosmology including sped up video of a radio telescope for the first 25 minutes.
The last ten minutes was a discussion of the future, what we may make of it.
All good stuff BUT what did he actually say about the universe and mathematics? What was his overall thesis? The Universe IS mathematics....Is that it?
Right, how does that work? What does that mean?
We can always buy his book of course (available at most reputable bookstores...)

Thanks for sitting through this on our behalf. I would not have had the patience.

This is why some people suspect Tegmark (whose real name is Shapiro but which he changed to his mother's maiden name, as Shapiros are two a penny on the US East Coast) is more of a showman and self-publicist than a serious metaphysician.

I posted Massimo's criticism of Tegmark/Shapiro in post 50. Peter Woit gives a thoroughly damning review of his ideas, as expresed in a recent book, here: https://www.math.columbia.edu/~woit/wordpress/?p=6551

These people seems to me to talk sense. Teggers, not so much.

 

[Pinball1970]

more of a showman and self-publicist than a serious metaphysician

Yes there is that because popular science sells books and videos.
Some cameras like these scientists and i am sure that their publicist jumped on that.
Brian Cox, Jim Al Kalili, Sabine Hossenfelder, Krauss, Turok, Kaku and personal favourite Sean Carroll.
Do not get me wrong, some of these guys are top notch published scientists but the persona and ideas they put out in pop sci books is nothing like what they discuss in conferences.
Tegmark's presentation is very see through,he said virtually nothing regarding his ideas. Are they even his ideas?
Feynman discussed this issue (see his 'onion' analogy)
Penrose too although he was a lot more humble, "we do not know why the universe seems to be governed by mathematics."
This idea is hardly original.

 

[Pinball1970]

I posted Massimo's criticism of Tegmark/Shapiro in post 50. Peter Woit gives a thoroughly damning review of his ideas, as expresed in a recent book, here: https://www.math.columbia.edu/~woit/wordpress/?p=6551

Thanks for the reference. I will check that out, Woit and Smolin have written some decent books and articles regarding string theory. Ten years or so on from those books regarding the experimental evidence, it is still lacking. W.r.t. beyond SM physics? Besides that little G2 wobble! Save that for another thread!

 

[exchemist]

Yes there is that because popular science sells books and videos.
Some cameras like these scientists and i am sure that their publicist jumped on that.
Brian Cox, Jim Al Kalili, Sabine Hossenfelder, Krauss, Turok, Kaku and personal favourite Sean Carroll.
Do not get me wrong, some of these guys are top notch published scientists but the persona and ideas they put out in pop sci books is nothing like what they discuss in conferences.
Tegmark's presentation is very see through,he said virtually nothing regarding his ideas. Are they even his ideas?
Feynman discussed this issue (see his 'onion' analogy)
Penrose too although he was a lot more humble, "we do not know why the universe seems to be governed by mathematics."
This idea is hardly original.

Ah but “governed by” mathematics, which is just a fancy way of saying it displays ordered behaviour, is not at all the same as saying it IS mathematics. That is where Teggers loses people like Pigliucci - and, at a much more humble level, people like me. It does not seem to follow at all and doesn’t even make sense, if one sees mathematics as a structure of quantitative logic, i.e. essentially abstract.

I like Sabine and I don’t know Sean Carroll. I’m rather suspicious of Cox, Al Khalili and especially Krauss, who is silly enough to dismiss philosophy, thereby showing a poor grasp of the foundations of his own subject.

 

[Sarkus]

Ah but “governed by” mathematics, which is just a fancy way of saying it displays ordered behaviour, is not at all the same as saying it IS mathematics. That is where Teggers loses people like Pigliucci - and, at a much more humble level, people like me. It does not seem to follow at all and doesn’t even make sense, if one sees mathematics as a structure of quantitative logic, i.e. essentially abstract.

I struggle with the "is mathematics" as well, but then sometimes I think it does sort of make some sense (in the way that a refelction in a hall of mirrors sort of looks like the person).
There's the whole "map v territory" category fallacy that it seems to be, or "model v reality" to put it in a more mathematical scenario. So my thinking about it is that the more detailed the mathematical model, the more closely the output reflects reality. This isn't novel. We all aspire, when we make such models - be they financial or otherwise - to make them as reflective of reality as possible. So, to mix analogies, at what point does a working model of a duck actually become a duck? How detailed must the model be to reflect reality perfectly? And if something is a perfect model of reality, then surely it ceases being a model at all, and actually is the reality that you're modelling, right?
So, my thinking is that if you can create a mathematical model that perfectly represents reality, then that mathematical model must be that reality. And hence reality could be considered mathematical. And that's my sticking point in this line of thinking (as woolly and unscientificly simple as I've tried to make it) - is it actually possible to create a mathematical model that perfectly reflects reality?
If you create a model of a duck that perectly looks, walks, smells, quacks, and does everything else a duck does and does them exactly like a duck... is it not a duck?

A similar line of thinking - and excuse the slight sidetrack - seems to be with regard the "Simulation Hypothesis", the idea that we, and our universe, are actually nothing more than a simulation being run in a (albeit rather powerful) computer. It has serious probabilistic arguments for it, but the point is not whether it's true or not, but that if true then it would also suggest that our universe is nothing more than a series of 0s and 1s, or whatever computational system the computer is running on. I.e. something akin to it being mathematical.
One could argue that with regard the MUH, the Simulation Hypothesis really just pushes the issue back a layer, but I thought it sufficiently worthy of comparison.
Anyhoo - just a side note.

 

[Write4U]

This is why some people suspect Tegmark (whose real name is Shapiro but which he changed to his mother's maiden name, as Shapiros are two a penny on the US East Coast) is more of a showman and self-publicist than a serious metaphysician.

I posted Massimo's criticism of Tegmark/Shapiro in post 50. Peter Woit gives a thoroughly damning review of his ideas, as expresed in a recent book, here: https://www.math.columbia.edu/~woit/wordpress/?p=6551

Interesting critique without a shred of hard science.
This is in one of the replies to the critique:

I mean, when modern physics points to fact that solid, physical matter is in fact vast amount of empty space linked together by interactions of profoundly small particles that seem to have an ephemeral existence all their own, is Platonism or realism about abstract structures and mathematical relations underlying the physical world really so outlandish? I agree it could probably never be tested, making it more philosophical than empirical, but do you think it a reasonable view?

I do.

By the way Frenkel’s book, Love and Math, is brilliant so far and its clear from the reading that he, along with a long list of other mathematicians, wholeheartedly embraces the Platonic view without the slightest bit of crackpot “mysticism.”

 

[exchemist]

I struggle with the "is mathematics" as well, but then sometimes I think it does sort of make some sense (in the way that a refelction in a hall of mirrors sort of looks like the person).
There's the whole "map v territory" category fallacy that it seems to be, or "model v reality" to put it in a more mathematical scenario. So my thinking about it is that the more detailed the mathematical model, the more closely the output reflects reality. This isn't novel. We all aspire, when we make such models - be they financial or otherwise - to make them as reflective of reality as possible. So, to mix analogies, at what point does a working model of a duck actually become a duck? How detailed must the model be to reflect reality perfectly? And if something is a perfect model of reality, then surely it ceases being a model at all, and actually is the reality that you're modelling, right?
So, my thinking is that if you can create a mathematical model that perfectly represents reality, then that mathematical model must be that reality. And hence reality could be considered mathematical. And that's my sticking point in this line of thinking (as woolly and unscientificly simple as I've tried to make it) - is it actually possible to create a mathematical model that perfectly reflects reality?
If you create a model of a duck that perectly looks, walks, smells, quacks, and does everything else a duck does and does them exactly like a duck... is it not a duck?

A similar line of thinking - and excuse the slight sidetrack - seems to be with regard the "Simulation Hypothesis", the idea that we, and our universe, are actually nothing more than a simulation being run in a (albeit rather powerful) computer. It has serious probabilistic arguments for it, but the point is not whether it's true or not, but that if true then it would also suggest that our universe is nothing more than a series of 0s and 1s, or whatever computational system the computer is running on. I.e. something akin to it being mathematical.
One could argue that with regard the MUH, the Simulation Hypothesis really just pushes the issue back a layer, but I thought it sufficiently worthy of comparison.
Anyhoo - just a side note.

Yes but this is where the error comes in, it seems to me. You can't build a mathematical model of something physical without first defining certain entities and their attributes, which you have to do in words. "An electron." "Energy." Momentum." etc.

None of these things can be defined by an equation without some words behind, to explain the physical concepts the equation relates together. This is where people can lose track of the physical nature of models in their worship of the maths.

 

[C C]

It is not difficult. I did not say "the universe is mathematics", that's Tegmark.

My personal POV is that the universe is a mathematical object, a pattern, and all universal functions are based on and guided by generic logical mathematical laws based on relational values and functions that humans have observed and been able to symbolize and codify. [...]

Mathematical universe hypothesis: Tegmark's MUH is the hypothesis that 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. Mathematical existence equals physical existence, and all structures that exist mathematically exist physically as well. Observers, including humans, are "self-aware substructures (SASs)". In any mathematical structure complex enough to contain such substructures, they "will subjectively perceive themselves as existing in a physically 'real' world".​

This seems redundant at first glance. What's the difference between declaring the universe a "physical structure" and declaring it a "mathematical structure"?

If the latter is completely disassociated from graphics or pictures, then "mathematical structure" might become radically different by consisting purely of technical description.

But that's the point where few, if anybody, can fathom what mathematical realists (in general) are claiming with respect to their abstract entities being "real", or how they exist. If they're not the redundancy of "physical forms" or spatial configurations, and they obviously can't be our artificially invented symbols (which are also impotent in terms of generating slash governing anything on their own).

All I can see that's left is some fuzzy idea that mathematical entities are nomological or immaterial "generative principles" that output and regulate phenomenal or physical "stuff". Thereby turning the universe into a continuing process rather than a complex, higher-dimensional physical structure where all its "developing" states co-exist.

Which seems to be an obscured way of mathematical realists claiming (without the directness) that the universe is a simulation. Where one configuration of the world is created and replaces the last one according to rules, maintaining inter-consistency of such states over time. But not a simulation residing within a computer (which would just be a fallacious repeat of the nature of this world). But riding atop whatever that weird, immutable realm of "mathematical" generative principles is whose only [supposed] evidence is its creative/governing effects on this universe (simulation, process).

Even advocates of philosophical presentism might dismiss that as a superfluous add-on, akin to the old-fashioned supernatural. Preferring instead that the order -- the "lawfulness" in their view of the universe as a process -- just happens or is "magically" maintained without an extraneous or underlying nomological domain (stratum of laws or generative principles -- whether dubbed mathematical or otherwise).
_