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Cosmology, Quantum Mechanics & Consciousness Message Board › What has maths got to do with science? (Arf!)

What has maths got to do with science? (Arf!)

A former member
Post #: 59
Wotcher everybody.
I did the decent thing and e-mailed Frances to suggest a proper meet up at the usual place. No response yet, but frankly it’s usually the aftermath in the pub that’s the best bit anyway. Andrew, I don’t think it is so arcane that it couldn’t be discussed over a pint and the great advantage is that when someone says something that requires a thesaurus and degree in extreme cleverology to decipher you can ask them to dumb it down in real time. Actually, if this comes off, I’m thinking of inventing exasperation sticks, basically a stick with a post-it on it that says ‘WHAT?’ on it, but don’t you think it is priceless that Ian thinks he is such a model of clarity he is translating Camilla for us? Sorry to tease, you are difficult to follow, but I at least think it is worth the effort.
So if I haven’t offended too many people I’d like to suggest we meet up on Tuesday 11th December at a venue to be argued about. I’m based in Ealing and there are some nice pubs just a few minutes walk from the Piccadilly line. Alternatively, my nephew runs pub not far from East Putney on the district line, but I’m open to suggestions. Please note, this invitation isn't restricted to people who contribute to the message board; you are all very welcome to come along.
A former member
Post #: 174
Yes indeed, much thanks Ian for the running translation in the previous thread. Beautifully crafted - I particularly appreciated the useful added insight that ‘instrumentalist’ can have a slightly pejorative connotation. It made me think of Andrew Mitchell, who had a conversation with a policeman in Downing Street recently. He should not have called the officer a pleb. Instead he should have called him an instrumentalist: a properly esoteric put-down that would never have made it to the news headlines. And then the Chief Whip would have kept his job! (With my full apologies for prolonging the attempted humour.)

Will, I would definitely support your proposed discussion but I cannot make it on that date. I suspect that very few people look at the message board these days so you might need to set the topic for a full MeetUp at a later date.
lan B.
user 10895495
London, GB
Post #: 273
OK. Interesting (?!)

Camilla: as indicated earlier we've had severe connection problems today and so trying to prioritise my time is difficult. I'll respond to your follow-up mail over on Andrew's "About this group" thread probably Monday.

Thanks for the glowing praise Andrew but I have to confess that it's unmerited. Instrumentalist as a general term within the philosophy of science needn't be pejorative. However the historical record within scientific theorisation and empirical discovery unequivocally demonstrates that instrumentalism has without exception always been mistaken! Ever since its somewhat over-subtle introduction by Catholic theologians around the 12th or 13th centuries it has been shown that Earth does orbit the Sun -- and not vice versa -- that matter is atomic, that our planet is billions of years old, and the only trenchant outlier remaining is, of course ..

.. the interpretation of quantum mechanics! (Which is precisely why -- and quite rightly - everyone continues to make such a fuss about it.)

The philosopher Baas van Fraassen is the current locus classicus for the instrumentalist perception of QM, true to his culturally Catholic roots, but then again as his website shows he's also a convinced cat-lover, and so I retain a soft spot for him nevertheless.

.. But what about Tegmark -- and Deutsch? .. and Ekert? .. and a large coterie of mathemetical platonist fellow-travellers? Does there exist any sort of rationally acceptable position in support of the thesis: "Its from Bits"?

What might someone be getting at in expressing such an incomprehensible thesis as "Numbers are the fundamental consistent of reality?" Do they perhaps mean to claim that any sufficiently sophisticated chemical analysis would reveal numbers as the ultimate sub-units? Or maybe we should consider some analytical tool with a little more clout. How about trying to find numbers by tuning the 27-km accelerator ring at CERN to higher energies?

Alright, clearly no-one (?) believes any of what I have just typed. What, then, are such advocates actually advocating?

Platonists are mathematical thinkers who are first and foremost not instrumentalists. They're instead staunch realists. The only problem is, they choose to apply their realism outside the true -- i.e. the physical sciences!

So I suggrest for the moment -- yet welcome dissenters -- that the interesting question devolves to:

"How should mathematics be interpreted? Is it merely an adjunct to sophisticated, entirely general problem-solving? Or does it have more subtle ontological (sorry Andrew) implications, lurking in the wings (as the platonists claim?"

My connection problems remain for the present. Hopefully I've provked a few raging bull responses waiting for my perusal on Monday, so that's all for now (although as you may anticipate I could churn forth for a month, and may well end up doing so.)





Camilla M.
user 7151822
London, GB
Post #: 42
I must be explicit - I don't trade in nasty asides. Therefore, I didn't mean anything pejorative in using the term instrumentalist, as usual it is a neutral term. Whatever Ian thinks that I feel about engineers, which more accurately reflects his own view, it is quite wrong in the context of discussions where I have indeed dissected the types of approach precisely to appreciate that engineers will often want to look at what works rather than at purely semantic matters. I wouldn't go so far as to say that instrumentalism has never won out, I think the instrumentalist / realist question is simply a long running very basic issue in philosophy of science which looks at its framework.

In terms of circling around whose approach we like on here, I'm far more interested in content than style. I can't understand why Andrew firstly says that he is alienated by both Ian and Camilla's writing, wouldn't want to join any in person discussion with us, then that the former seems well crafted. What am I to make of this: Which is it?

I second most of Ian's questions about instrumentalism, specifically what comes of conflating numbers with reality. As I told him, I was drafting my own response but put it to bed to have a rethink as I could anticipate the kinds of answers I would get in keeping with the instrumentalist stance. Things like, when I would ask about the Higgs Boson search, I'd probably be told, in puzzlement, that the frequency of the signature range shows that a reading is repeatable in keeping with scientific inquiry and that that just IS proof of existence. This is why realism remains a vexed question.

Somehow, I wanted to ask, well if reality is number representation and vice versa, and experiments are made of statistical aggregates of numbers ultimately, then what is material structure and form precisely? [As opposed to any mathematical treatment of the same.] It is not commensurate with differences in the original numbers' layout specifically but seems to be of a second or n order remove, in other words much as realists conceive of maths as meta commentary in the first place!

If this is the case, all we've done is renamed reality as one set of numbers then allowed structure, form and other useful descriptions and predictions to be given exactly in other mathematical formalisation as well as less precisely in language. That looks similar to a somewhat deprived form of realism, again. The question then being how nestedly rich the initial mathematical description of real things as numbers need be? In order to account for all the properties we see in distinct material. It all looks like knowledge about an external world rather than being the world in itself.

Could we though design a reverse experiment where numbers masquerading as they must as experimental equipment produce both predictable of course further numbers - experiments generally produce numerical quantities along measurable dimensions which seems an exchange between number and whatever material is anyway - but in this case we would then want them instead to bypass the purely mathematical treatment and be seen to emerge directly as materials?? What extra performance on a usual numerical output is required to convert straight to the correct identical actual materials we see in the everyday world? [Not just any old 3D printer plastics in shapes out - this would be a truly magic 3D-making abacus.]


Camilla M.
user 7151822
London, GB
Post #: 43
Further, a middle paragraph above almost plays into instrumentalism, that is how hard it is to open up the discussion properly without immediately causing confusion. When I ask about structure and form, I mean in actual material terms the difference between properties of things in the usual reductionist terminology. Such as water and ice sharing the basic properties of hydrogen and oxygen electrically-bonded together. This would need to be included not merely in the basic mathematical description, which we can already offer but in some other way that makes it the actual substance, an actual invariable state of affairs (concensus propositional ontology,) not at one remove an epistemic description of it which can focus on various aspects. There is a difference between those multitudes of measurable observables, that we may sift or ignore, and our senses being confronted by any instances of irrefutable unchanging actualisation.
A former member
Post #: 175
I thought this thread was about setting up a meeting, but Ian and Camilla want to get on with a discussion. OK, then here are my thoughts on the above.

I detect a straw man in the preceding messages about numbers at the root of reality. If you browse the Tegmark paper that I mentioned you will find no suggestion that reality is to be equated with number representation. His actual words are: Our external physical reality is a mathematical structure. And he defines in precise detail what he means by a mathematical structure, which boils down in the most general sense to ‘abstract entities with relations between them’. It is normal to say that maths is founded on numbers but that’s not accurate. Maths is what you get when you apply the rules of logic to some axiomatic assumptions. Numbers come from one set of axioms, geometry from another set. Remove some axioms and you arrive at a more primitive and generalised maths. For example, loosely speaking, if you remove the metric from geometry then you get topology. If you remove the concept of proximity or neighbourhood from topology then you get set theory. If you remove the axioms of set theory there won’t be much left – except for the Boolean logic that underpins the edifice. Logic is itself a branch of mathematics with its own set of axioms. Numbers are not the starting point, they can be regarded as a structure built on set theory but built in a different direction from topology and geometry. So we can discard that amusing idea that the LHC will eventually uncover, at the base of everything, ....... numbers.

Anyway, back to Tegmark. To do him a disservice and omit all his arguments, he is basically saying that the universe ultimately consists of entities with relations between them, just like maths. For example, if and when we discover the ‘theory of everything’ then the mathematical symmetries of the fundamental structure of the universe will necessarily correspond with the laws of physics. The principle of symmetry is very important in both fundamental physics and maths. Planet Earth is a most complicated place but: simplify to an idealised sphere and there you have a perfect symmetrical form. Symmetry accompanies simplification. The symmetry of a sphere is described precisely in terms of a simple mathematical structure. So maths accompanies the reduction of physics on the journey down to its fundamentals.

Tegmark argues in this sort of way that the universe fundamentally looks exactly like maths, which perhaps is not too outrageous, but he then jumps to his hypothesis that ‘the universe is maths’. That’s a big jump. Its from bits indeed. We don’t need to make that jump. Though, if the universe is not maths then what exactly is the difference? What distinguishes reality from an exact model of reality? As Stephen Hawking once asked: ‘Why does the universe go to all the bother of existing? What is it that breathes fire into the equations and makes a universe for them to describe? Good questions indeed!

It is plausible that there are one or more levels of reality below our known physics, each emergent from the one below, just like the levels of emergence that we see in classical physics, chemistry and biology. To descend below classical physics, from one level of reality to the next one beneath, is much like stripping away the axioms of maths, moving from the developed specific to the more primitive generality. Einstein deftly removed Newton’s axiom of absolute time. At the most primitive levels, perhaps even logic would not hold. Logic itself could be an emergent feature. Indeed quantum foam is postulated for the structure of spacetime at the smallest Planck scale: that sounds like a structure where the metric is disappearing from geometry and maybe even logic has no role.

I quite like the recent Penrose idea of cosmological cycles of time. An ingenious and key aspect of the idea is the complete loss of metric for time and space during interim states of the universe when it contains only radiation and no matter. That would be a reduced state of physics prior to the Big Bang, analogous to the metric axiom being removed from geometry.

Changing perspective slightly, how would you design a universe from scratch? What would you use for its constituent stuff? Numbers? No, we have dismissed that. Quantum wave functions? No, they don’t look fundamental enough - how on Earth could you breathe fire into Schrodinger’s equation? What about binary digits, like 0 and 1, or on/off? That looks much better: a binary representation is the simplest we can imagine. And what is the most fundamental aspect of a universe? The state of existence. To be or not to be, that is the answer. Let’s design our universe out of binary states of exist / not exist. And then the second most fundamental idea, that of change, is naturally created by allowing fundamental states to flip between ‘exist =1’ and ‘not exist = 0’. From change will emerge time. So let’s seed the new universe with a couple of 0/1 states and let them propagate. No spacetime, just a dimensionless set of states, just maths really. See what happens. Program it on a quantum super-computer from the 22nd century. See what emerges. Maybe some quantum foam! The makings of a real universe?

So maybe the universe is basically numbers after all, in the form of binary states. The Beckenstein bound on information content of a spatial volume points that way. Convergence between the foundations of mathematics and the foundations of physics: that’s why maths is relevant.
Camilla M.
user 7151822
London, GB
Post #: 44
How lovely to have a detailed exposition from you, Andrew! I've digested it as best I can, well aware that the practising physicists and mathematicians amongst us will have the head start on metrics and the mathematical overview. My questions and comments are very naive, largely for clarification, read them in the spirit of inquiry and not to take precedence over more incisive ones.

‘abstract entities with relations between them’.

Sounds like the philosophically structural view of reality.

Contrasting with:

Let’s design our universe out of binary states of exist / not exist.

Which might be the most basic form of an essentialist view (intrinsic-property based.)

I've had a book on my shelf for a decade or so called Two State Universe by Angelo A. Molinaro, who espouses something similar to the binary formulation, if it's of any interest.

Symmetry accompanies simplification.
The symmetry of a sphere is described precisely in terms of a simple mathematical structure.

The structural position then would seem to reduce the precision of information about its components whilst offering a higher level (possibly entropic) view?

It is plausible that there are one or more levels of reality below our known physics, each emergent from the one below, just like the levels of emergence that we see in classical physics, chemistry and biology. To descend below classical physics, from one level of reality to the next one beneath, is much like stripping away the axioms of maths, moving from the developed specific to the more primitive generality. Einstein deftly removed Newton’s axiom of absolute time. At the most primitive levels, perhaps even logic would not hold. Logic itself could be an emergent feature.


The materialist's view does takes logic and later number theory as arising or 'emerging' practically from perception of the world, rather than preceding it. That is my question in response to Will's question - whether we push logic from its own necessity into having to be considered priorly existent, despite it's being perhaps also thought of as an empirical matter.

quantum foam is postulated for the structure of spacetime at the smallest Planck scale: that sounds like a structure where the metric is disappearing from geometry and maybe even logic has no role.

Logic being about propositions, the primary form being identity, where p either = q or ¬q, then binary on/off existence (of what? in property rather than coordinate terms, I suppose) would seem to be basically logical?

With no metric, can we get symmetry breaking off the ground?
Each on/off posit would either be indistinguishable or 'superimpose' if that's the right idea: would there even be separation as in set theory at this stage?

I had a look at the Bekenstein bound re upper limit to entropy, a finite value for a finite space and energy universe. I'd come across this from the holographic principle promoted by Susskind.

That's enough tonight.

(Just changed p => q)
















A former member
Post #: 176
Thanks Camilla. I bet Ian would like to interface with the philosophy framework so I will defer to him on those points, where I would struggle. That leaves me with a few quick comments on how to design a universe (!!):

That book can stay on your bookshelf! Probably not the same idea at all, and not a book of any note, apparently.

“Logic being about propositions, the primary form being identity, where p either = q or ¬q, then binary on/off existence (of what? in property rather than coordinate terms, I suppose) would seem to be basically logical?”

Yes, though I’m presently allowing the theme of reductionism to take my imagination to a place beyond logic, where you just have binary bits or states of ‘to be or not to be’, these bits conveying the irreducible essence of reality and existence. A book on my shelf is ‘A new kind of science’ by Stephen Wolfram. It’s a huge tome and perhaps we can forgive Wolfram for his pompous title, as he shut himself away for a solid ten years to research and write the book. His theme is to illustrate how the known laws of physics can seem to emerge from binary bits chasing round in a computer simulation, very much like Conway’s Game of Life. (PS wouldn’t it be great if Shakespeare’s most famous line turns out to be the long-awaited and ultimate scientific Theory of Everything?!)

“With no metric, can we get symmetry breaking off the ground? “

If the universe kicks off with no structure, in a state of maximum symmetry and minimum entropy, then that absolute perfection at outset would be eternal perfection and we would not be here. It would surely be sufficient for the tiniest imperfection in the initial symmetry to enable the emergence of more asymmetry later and with a potentially interesting universe eventually to follow.

“Each on/off posit would either be indistinguishable or 'superimpose' if that's the right idea: would there even be separation as in set theory at this stage?”

I get your picture. I am indeed suggesting a picture with no space in which to house these elements of a universe. Space itself would be a later emergent feature, and Wolfram has evidence of how this could work. But yes, these states need to be distinguishable. And yes again, this germ of an idea might lead to a workable basis for quantum superposition. (But then again maybe not. I would not dare to adopt a position on this!)
lan B.
user 10895495
London, GB
Post #: 274
I must be explicit - I don't trade in nasty asides. Therefore, I didn't mean anything pejorative in using the term instrumentalist, as usual it is a neutral term. Whatever Ian thinks that I feel about engineers, which more accurately reflects his own view, it is quite wrong in the context of discussions where I have indeed dissected the types of approach precisely to appreciate that engineers will often want to look at what works rather than at purely semantic matters. I wouldn't go so far as to say that instrumentalism has never won out, I think the instrumentalist / realist question is simply a long running very basic issue in philosophy of science which looks at its framework. .

I wasn’t imputing to you any designs of nastiness, Camilla. I think what actually happened is that Andrew found that the notion of someone who customarily expresses themselves in philosophical terminology could come across as some sort of inadvertent exposer of murkily and skilfully hidden “philosophical dirty laundry” too delicious to resist! ( .. But I could of course – as in any other conjecture – be entirely mistaken about this.)

It does indeed seem unsettling that the existence of the Higgs Boson needs seemingly to be validated on purely statistical grounds, but the problem with the ATLAS detector and others of the modern generation of “particle”-track detectors is precisely that there are at least tens of millions of interactions within the “field of viewt any one time. The situation is definitively not like that which obtained when we were children, with nice, clear, unequivocally curly tracks coiling in one sense within an ambient magnetic field – indicating the clear presence of an electron – and another exactly symmetrical “anti-curling”, mirror-image track clearly betraying the presence of a positron (and so on).

Maybe I would be upsetting the apple cart by reminiscing about a talk given by a CERN researcher at the Kings College (London) Maxwell Society > 5 years ago stating that what was then needed was simply a hugely greater number of runs in order to furnish the needed statistical base. (At that time they wanted 8 dex, and they only had 5 with confidence, given the data and the then-long time needed between runs. The Higgs’ mass was also said to be well-constrained both from above and below to a range centred around 115 GeV, yet recently we see an entirely different range being advocated if I remember rightly. I need to check the details but we’ve only just got back online!)


Somehow, I wanted to ask, well if reality is number representation and vice versa, and experiments are made of statistical aggregates of numbers ultimately, then what is material structure and form precisely? [As opposed to any mathematical treatment of the same.] It is not commensurate with differences in the original numbers' layout specifically but seems to be of a second or n order remove, in other words much as realists conceive of maths as meta commentary in the first place!

If this is the case, all we've done is renamed reality as one set of numbers then allowed structure, form and other useful descriptions and predictions to be given exactly in other mathematical formalisation as well as less precisely in language. That looks similar to a somewhat deprived form of realism, again. The question then being how nestedly rich the initial mathematical description of real things as numbers need be? In order to account for all the properties we see in distinct material. It all looks like knowledge about an external world rather than being the world in itself.


Yes I agree that mathematics is an epistemic apparatus rather than “something intrinsically “there” “, which is precisely the reason that I continue to resist mathematical Platonism in its fully-fledged form. It seems to me that it is an ineradicable property of “existence” that the existent can somehow be sensed – no matter how indirectly or in however long-chain inferentially derived a manner. Even if CERN experimenters could somehow succeed in hacking fundamental “particles” to bits at ever-higher collision energies, one remains nevertheless scratching one’s head at the vexed question of what, then, numbers (or any other mathematical expressions for that matter) would actually look like.

Could we though design a reverse experiment where numbers masquerading as they must as experimental equipment produce both predictable of course further numbers - experiments generally produce numerical quantities along measurable dimensions which seems an exchange between number and whatever material is anyway - but in this case we would then want them instead to bypass the purely mathematical treatment and be seen to emerge directly as materials?? What extra performance on a usual numerical output is required to convert straight to the correct identical actual materials we see in the everyday world? [Not just any old 3D printer plastics in shapes out - this would be a truly magic 3D-making abacus.]


Yes, well you’ve pre-stated in periphrasis the question that I’ve asked just above!


Further, a middle paragraph above almost plays into instrumentalism, that is how hard it is to open up the discussion properly without immediately causing confusion. When I ask about structure and form, I mean in actual material terms the difference between properties of things in the usual reductionist terminology. Such as water and ice sharing the basic properties of hydrogen and oxygen electrically-bonded together. This would need to be included not merely in the basic mathematical description, which we can already offer but in some other way that makes it the actual substance, an actual invariable state of affairs (concensus propositional ontology,) not at one remove an epistemic description of it which can focus on various aspects. There is a difference between those multitudes of measurable observables, that we may sift or ignore, and our senses being confronted by any instances of irrefutable unchanging actualisation.

Yes we’re clearly thinking along the same lines Camilla. A Platonist opponent could of course observe that we are apparently without further criticism presupposing the argument in favour of “material entities”, and if we’re allowed to make that move then the argument becomes doomed right at the outset be settled in our favour. I feel bound to reiterate though that I somehow think that what it means “to exist” is for the existent to be potentially sensed, whether directly to the naked eye, ear, nose or skin surface or instead and more usually within scientific research via a subtly long chain of instrumental or chemical detection in conjunction with trusted and prior-established background assumptions/knowledge, this chain of reasoning in itself being largely mediated with the mandatory assistance of calculations!

[ .. continued .. ]

lan B.
user 10895495
London, GB
Post #: 275
[ .. Continued .. ]

One of the (less significant IMHO) bugbears about QM is the fact that the actual status of “materialism” has been radically changed since the 1920s, which is why the position held by folks like myself these days now flies under the critically much more acceptable banner of “physicalism” – i.e. “whatever contemporary physics says is the case (subject to the essential caveat that either the predictions of some respected and pedagogically speaking universally accepted theory such as GR or QM have already been long and satisfactorily confirmed, or that the density of already-made observations established “under the aegis” of some such over-arching theory is such that “there ought to be” states of empirical affairs which unfortunately haven’t quite yet, embarrassingly, been deteced; gravitational waves within the theoretical province of GR itself being a notable current example).
).

[Andrew]: I thought this thread was about setting up a meeting, but Ian and Camilla want to get on with a discussion. OK, then here are my thoughts on the above.

I have no objections against meeting in person. My understanding had been that you found this difficult since you apportion your time sparingly between London and Bristol (?) but for the time being there’s no reason why we should not better try to understand each others’ position.

I detect a straw man in the preceding messages about numbers at the root of reality. If you browse the Tegmark paper that I mentioned you will find no suggestion that reality is to be equated with number representation. His actual words are: Our external physical reality is a mathematical structure.

No quarrel except to modify as follows: External physical reality is isomorphic to (some set of) mathematical structures.. Funnily enough, at yesterday’s fortnightly meeting of the Aristotelian Society in Senate House, Malet Street, London W1 – on the putative ontological connection between correct theories within mathematical physics and their aesthetic appeal – I pointed out to a largely humanist-leaning audience the fact that the spatial extension, the duration of events, the mass and the velocity of systems as judged by observers observing between inertial frames (within the apparatus of SR) exemplify the structure of a mathematical group, with identities and symmetries, and that this interesting and possibly pleasing property is not enjoyed by its predecessor, Newtonian kinematics (other than in the somewhat empty sense that the product of the value of the gamma coefficient and its associated intrinsic algebraic numerator coefficient must always, trivially and uninformatively, be equal to one!)!)

And he defines in precise detail what he means by a mathematical structure, which boils down in the most general sense to ‘abstract entities with relations between them’. It is normal to say that maths is founded on numbers but that’s not accurate.

I agree. (I was being provocatively facetious.).)

Maths is what you get when you apply the rules of logic to some axiomatic assumptions.

In a sense you’re right, but rather I would (more generally) say that problem-solving – even when playing games such as Cluedo or Monopoly – is what you get when you apply the rules of logic to some axiomatic assumptions.

(The interesting question is: can mathematics by in-principle extension subsume any game whatsoever ?– and I’m not simply discussing the already well-understood payoff matrices as encountered within the mathematically strictly defined Game Theory.)
.)

Numbers come from one set of axioms, geometry from another set. Remove some axioms and you arrive at a more primitive and generalised maths. For example, loosely speaking, if you remove the metric from geometry then you get topology. If you remove the concept of proximity or neighbourhood from topology then you get set theory. If you remove the axioms of set theory there won’t be much left – except for the Boolean logic that underpins the edifice.

.. Although I recall from the ‘90s – again at Kings College Philosophy of Science seminars – Moshe Machover expressing his own admiration for and bewilderment at Lattice Theory, which he then said was able to underpin Set Theory in a way which could disable the classical paradoxes, and also of course that its scope is, somehow, more general. (To my shame I have never checked it out. Quite clearly even he – genius that he was; he knew that GCHQ were actually running the Wittgensteinian experiment of reading people’s verbal thoughts, and said so; well done and for once honest, that man! – found the maths daunting.).)

Logic is itself a branch of mathematics with its own set of axioms.

Indeed so! Or rather, “logic” is in fact an entire infinite family of axiomatically constrained decision procedures, with the familiar and very sensible Aristotelian form being just one species! ( .. And the ultimate judge from our point of view of the utility of applying any of the others!)!)

Numbers are not the starting point, they can be regarded as a structure built on set theory but built in a different direction from topology and geometry. So we can discard that amusing idea that the LHC will eventually uncover, at the base of everything, ....... numbers.



Anyway, back to Tegmark. To do him a disservice and omit all his arguments, he is basically saying that the universe ultimately consists of entities with relations between them, just like maths.

No scientist or philosophical physicalist would disagree with that. What is interesting is that the decomposition of these entities and relations on the basis of phenomenological inspection alone need not lead to any unique decomposition! .. Popper again: A literal infinity of theories can satisfy any (finite?) data set.

(“Subtle is the Lord!”)
”)

For example, if and when we discover the ‘theory of everything’ then the mathematical symmetries of the fundamental structure of the universe will necessarily correspond with the laws of physics.

Yes .. in the sense that the mathematical map of reality at its most general, axiomatically expressed form would, necessarily, be obliged to conform to our empirical findings, using big pieces of engineered, physical equipment, in the same way that the latest version of some London street atlas is obliged to conform to the broadly useful-from-the pedestrian-perspective features of the actual streets of London, to be found as one travels along them.

Unfortunately, however, the map is not the territory! (The Platonists’ fundamental error, I fear!)
!)

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