Tegmark's Mathematical Universe

Henk,

You refer to mathematics, algebra, logic and set theory as though you think mathematics can be taken for granted: something that is assumed to exist that requires no further explanation. But surely "the right starting point" is to ask what mathematics is, and where it fits into the scheme of things? I think 4 points about mathematics are relevant:

First there is the question of a Platonic realm. I start from the assumption that there is no Platonic realm - that is, I think everything exists within this universe including mathematics: mathematics is part of the universe - it is not external to the universe. Unfortunately, even physicists who profess to not believe in a Platonic realm seem to treat their mathematical equations (laws of nature) as though they were external to the universe i.e. things unable to be explained, or things not requiring explanation in the context of this universe.

Second, I assume that mathematics arises out of the reality of living things: mathematics started out as, and continues to be, a system of symbolic representation. We are made out of the underlying reality of the universe; we are not separate from this underlying reality: I consider that mathematics could only make sense to us if and only if the basic elements /symbols of mathematics faithfully represented corresponding basic elements and relationships in the underlying reality. So there was no mathematics in the early universe - there was only the underlying reality, much of which we can represent using symbols*. These symbols include "+ - ﾃｷ ﾃ--" and "=", which presumably represent aspects of the underlying reality in this universe, not things that exist in a Platonic realm. These symbols, alone or as part of "law of nature" mathematical equations, seemingly represent aspects of reality that are not measurable, whose existence can only be inferred.

Third, while the basic elements/symbols of maths can be utilized to represent the underlying reality (e.g. as laws of nature), mathematicians or physicists can also creatively assemble mathematical symbols to represent artificial realities: this is similar to the way words in a story, or colour and line in a painting, can be used to represent actual experienced reality or to represent imagined, artificial realities**.

Fourth, the above views might lend support to a view that numbers found in nature, instead of being seen as mysterious Platonic objects, might be seen as being constructed out of the same "component parts" as laws of nature. E.g. if we say that "a + bc = d" represents a law of nature, then "(d +d + d)/d" represents a number similarly constructed. This view implies that that numbers found in nature have a hidden structure, but a structure that may be impossible to determine: there are many possible ways to construct the number 3 using mathematical symbols.

Lorraine

* But seemingly not all aspects of the underlying reality are able to be represented with mathematical symbols e.g. the precise "random" outcomes of quantum events are seemingly not representable as one side of a mathematical equation.

**This view would indicate that mathematics is both discovered AND invented: the type of symbols used in mathematical equations, and the form of a mathematical equation, represent discoveries about aspects of the nature of reality; but mathematics is invented in the sense that it is a creative reworking using these types of symbols.

Marcel,

You say "Physics is about our relationship with this universe" and "the universe has no need for measurements", but what do "relationship" and "measurement" mean in the context of these statements? You mention numbers, but what is a number? You say "the rule of non-contradiction is definitely the right one, as the starting rule of impossibility" - but where is the evidence that such a rule really exists - does it exist in a Platonic realm?

Lorraine

Hi Marcel and Lorraine,

I agree, physics is in part about our relations, as observers, with the observed universe. It is also in part about the relations of the material universe with itself as it is those relations that provide the forces for continual change (including the production of information that we are able to receive and interpret).

I like what was said about non contradiction. However contradiction can also just be the product of looking at things in a particular way, an opinion. Like the impossible triangle optical illusion. Or a flower that is patterned and not patterned, depending upon whether seen by a human or a bee. It is so with the incompatibility of QM and Einstein's relativity IMHO. Contradiction may be indicating a wrong idea and -or- a wrong or limited perspective of the observer; (To do with lack of complete information). Both are important in illusionists' work.

Lorraine,

IMHO the key to understanding the universe is a*b, the operation of multiplication between two algebraic elements. One cannot divorce algebra from mathematics, and it is *algebra* which defines the operation of multiplication between algebraic elements, one of three required operations with element addition and element multiplication by a scalar.

The operation of multiplication is not singularly defined, since there are many different algebras, each with their own definition. So there is a choice to be made on which algebra most succinctly matches up to physical reality. The other two required operations for algebra are commonly defined for all algebras, which is why * is so important.

In my essay for the last FQXi contest, The Algebra of Everything, I present a brief picture about why physicists need to take more notice of Octonion Algebra. The variability of how one may define * within Octonion Algebra and the realization it must impose no variability on the equations of physical reality *demands* Electrodynamics is the way it is, and provides sufficient dimensionality to include a potential theory of Gravitation in a natural coexistent way when the proper generally covariant definition for differentiation (provided) is used. Take a look if you have not already.

Rick

Dear Marcel,

you said .....

"Now, logic is taken here as having its full meaning. First it describes a valid operation on the substance like addition, substitution etc. .... Secondly, logic describes a spontaneous becoming i.e. the logical spontaneous outcome of a given state or situation. This spontaneous logical operation is the CAUSE. We have now recovered both the SUBSTANCE and CAUSE of the foundation of metaphysics. After some work, you should find out that there can only be one single type of logical cause in the universe. And this cause HAS to be built into a property of the substance; it can`t be otherwise. And this substance and this cause are....."

You aimed exactly right.

This substance and this cause are random numbers and imperative (only possible ones) logical operations on them. Yep, That is all to my theory which recreates all the physics we know.

http://www.qsa.netne.net

more detailed explanation are in posts 20,25,43 in this thread.

http://cosmoquest.org/forum/showthread.php?140913-%93Reality-is-nothing-but-a-mathematical-structure-literally%94

Lorraine (feb 28, 00:43),

Sorry, because of the misunderstanding; I was a bit vague. And to rectify: I don't take mathematics for granted. In fact, throw away all your books about mathematics because they are largely useless if you want to explore the mathematical universe. Most mathematics is some kind of applied mathematics. You need foundational mathematics or meta-mathematics.

[First] The Platonic realm was a fascinating new hypothesis about the nature of reality some 2500 years ago. Like string theories... (nice but at the long last useless because there are too much assumptions)

[Second] Nature forms atoms, quantum fields, chemistry elements and... humans too. We - humans - do not form nature. So there was math, there is math and there will always be math. With or without humans. (The laws of) Mathematics is the structure of the universe. But the "fabric" of the universe is unknown and will stay unknown. It is impossible to detect or deduce what is everywhere the same (we can only detect or deduce differences).

[Third] Look outside your window. Probably you see a house at the other side of the street. That is composite geometry (macro cosmos). And in the micro cosmos everything is geometry too (but on a fundamental non-composite level). So, no symbols, no mathematical notation system ( a sigh of relief ;-).

[Fourth] Numbers are representing objects (phenomena) and not vice versa. So there are no existential numbers to be found within the mathematical universe. Within the mathematical universe numbers represent mathematical objects and mathematical proportions. Just like we use to do in physics.

And the right starting point: you have to analyse the scope of reality (what is real and what isn't real). See the reply of Georgina Perry (feb 28, 09:03): you have to choose the right perspective as observer and explorer.

Henk

Hi all,

First, I think you are making a big deal out of nothing. People believe in that mathematical facts are true, the only controversy is why. Even that, most philosophers of mathematics will tell you that there is no need for the why, it is obviously just is.

Second, we deduce facts that we believe in, even though sometimes we don't actually touch or see them. We fully believe in the electron even though nobody has actually seen one. Also, we have not seen an actual circle either; its curved line width goes to zero. So fundamentally there is no barrier to deduce the ontology of reality.

Third, we use symbols to represent things like a circle. But again if no humans existed a circle will be there.

Forth, we do use objects to represent mathematical objects. Do you remember when you counted numbers with your fingers when you were a little kid? That's how we conceptualize mathematical facts with our brain.

Sure, clever conjectures by Tegmark and others have considered the possibility, but there has not been any credible theory that really ties reality to a mathematical realm, not until my theory that is.

Mathematics created our reality, and we are the living proof in the platonic existence of reality.

See my reply to Marcel.

Henk,

In [Second] you say "(The laws of) Mathematics is the structure of the universe". But the "random" outcomes from quantum events seemingly indicate that the reality that can be represented by mathematics is only part of the story. These random outcomes cannot precisely be predicted i.e. they cannot be represented as a (law of nature) mathematical equation. Mathematics is seemingly useless in predicting the precise individual outcomes of quantum events, so I don't think you can make an unqualified claim that "(The laws of) Mathematics is the structure of the universe".

You conclude: "you have to analyse the scope of reality (what is real and what isn't real) ". So, looking at the reality that CAN be represented by law of nature mathematical equations, we can assume that, thanks to the work of physicists, law of nature mathematical equations represent what is real, what actually exists within the universe.

What I'm trying to say is not necessarily about the precise details of individual law of nature mathematical equations: I'm saying that the form of a mathematical equation represents something about the nature of reality. The symbols "+ - ﾃｷ ﾃ--" and "=" are necessary parts of law of nature equations. My main point is that, just like the symbols "E" and "m" (representing information about energy and mass respectively) represent acknowledged aspects of reality, the symbols "+ - ﾃｷ ﾃ--" and "=" in these equations represent evidence of unacknowledged aspects of reality.

The symbols "+ - ﾃｷ ﾃ--" and "=" seemingly represent aspects of reality that are not measurable, whose existence can only be inferred. Science does not acknowledge that there are aspects of reality that are not measurable. If there is no Platonic realm, then these symbols represent fundamental aspects of reality that are part of this universe. Surely physicists cannot form a correct picture of reality unless they acknowledge, and attempt to analyse ALL aspects of fundamental reality, not just those like energy and mass that can be both measured and represented.

Lorraine

Lorraine (march 2, 11:37),

I understand and agree with your explanation about the potential relationship between mathematical operations (algebra) and the laws of physics. But a big part of the problem is the impossibility to understand the mathematical reality by using the descriptions (equations) of phenomena in physics. And that's because physical phenomena do not represent the primary level of reality. So you cannot build yourself an image of this underlying reality (and that's the problem in quantum field theory too).

Your example: E = mc2.

This equation means: the invariant mass of a system is the total energy divided by the speed of light squared. Now the meaning of this equation "within the mathematical reality".

E is the amount of moved area into the volume.

And... everything is clear? Of course not. This is another kind of incomprehensible "speech".

So trying to explore the mathematical reality has nothing to do with translating applied mathematics and theoretical physics into this mathematical model of reality. It is just the other way: exploring the mathematical reality will solve all those incomprehensible theories that are composed by physicists and mathematicians.

We have to accept the fact that empiric science is founded by the process of analysing (composed) phenomena. These phenomena are formed by one underlying structure of reality (that's the conviction of most scientist). So, we cannot explore reality by comparing physics and mathematics. That's not the way. You have to "build" reality from the bottom up with the use of mathematics. And of course: that's (some kind of) logic and set theory (the proposed foundations of mathematics).

Physicists have tried to form a correct picture of reality by attempting to analyse all aspects of fundamental reality, but they didn't succeed (e.g. string theory and super symmetry).

Henk

Lorraine,

..Relationship with the universe... Everything we know we learned from our interaction with the universe in the capacity of an observer. If you think or look at two particles about to interact, you are an observer. In that capacity you may think you are not involved in what is observed. This is wrong in many ways. You are a third entity but you are not part of the equation; what are you?. And yet, what you look at is highly transformed by using sense and brain. The point of view of the observer, a third party, is in no way a part of the universe. How could the universe be made of a point of view if we only showed up recently in this universe? To understand the universe you have to leave your special position of observer and take the place of one of the two particles. Now you may be part of the action in the capacity of an actor. ... In such a place there is no numbers because you don't "know" what is around you or away... You only feel whatever where you are ... no numbers... no equation .... You exists, that's all. And existence is the most basic property to consider.

Georgina,

Non-contradiction (NC) is the logical basis of everything, maths ,science . .. Illusion is perception, as is our reality. GR and QM are both right as different points of view (truth systems) and as such can not be reconciled. Consider that the point of view of each actually look at the same thing. Then, only in the actual subject of their observation can there be unity i.e. the underlying reality.

Qsa,

Although a unique substance makes for an incredibly efficient use of numbers they must support an actual substance that exists by itself. The universe is not made of numbers. There must be a logical primer in order to get everything moving. Numbers are for people who wants to know "how much". The universe only need a "why" to get moving. (I will check your site -thanks)

Lorraine Ford wrote on Feb. 28, 2013 @ 00:43 GMT

You explain in more details your case/theory. Yes, maths being born out of logic do relate to actual logical operations the universe performs in the underlying reality. But all the metric aspect is for your own intellect - need to know. The universe only needs a logical reason to evolve. We may sum up and quantify these logical reasons in one place in order to come up with a "law of nature" that describes or match our "observations". But Nature does not follow the law; it follows the logic.

Rick,

The question could be: Does nature effect a *? How does it do that with a substance? What is the greater naturalness of octonions? Better description or better understanding?

Henk,

"But the "fabric" of the universe is unknown and will stay unknown." Well, we have learned over the years to regard "never" statements with a certain smile... We can deduce what the substance is. The (scarry) part is: do we really want to? Are we ready and mature enough to do so? Questions for which we really don't want the answer are typically not being asked.

qsa qsa replied on Mar. 1, 2013 @ 00:24 GMT

qsa,

"But again if no humans existed a circle will be there." Nope. WE make the circle full. By perception we integrate and consider a bunch of points and make the circle. AS an observer you take the picture and make it as a circle. As an actor you would be one of those points not knowing anything beyond just you.

Physics tells you the same thing. Because the speed of light has a limit, these points can't interact(1) with each other simultaneously i.e each point is in fact away in time from each other. Therefore, no two points of this "circle" are at the same moment! Your privileged position as an observer may allow you to see them all at once in a (same) moment of perception. But they are not at the same moment to form a "circle". They are a sequence of points accross time all having A same time distance to each points of (central) line. (Otherwise, they could form a line drawn on the surface of a sphere) .

(1) To involve the "interaction" between points detaches you from a spectators point of view and gives you an operational point of view.

Marcel,

Georgina,

Well, I don't buy the wave collapse thing. Try to understand a 100km wavelength radio wave "collapsing" on an antenna. It rather induces current in the antenna as it comes in instead of collapsing onto it. Can you figure out a 100km wavelength radio wave as a "photon"? Mighty big particle! "photons" are waves, not particles. More like solitons waves.

Marcel

Hi Marcel,

I'm not imagining an actual collapse either. In my own explanatory framework the equivalent of the hypothetical wave function collapse is just a switching from thinking about an independently existing "reservoir" of potential sensory data and all of the possibilities encoded within it to thinking about the one observed output from processing of a selection of the potential data.

Hello Marcel, Georgina, qsa, Rick and Henk.

Re "Relationship with the universe... Everything we know we learned from our interaction with the universe in the capacity of an observer...You exists, that's all. And existence is the most basic property to consider" (Marcel, 2 March, 20:20):

I argue (e.g. in my essay and posts in the 2012 FQXi essay competition) that apprehended information is subjective experience*. I also contend that information is physical and "the physical" is information. So I agree with you, Georgina and Marcel, about the primary importance of subjective experience/ information in every aspect of the universe.

But here I am interested in examining a different issue related to the topic of Tegmark's Mathematical Universe: the issue of how to interpret the components of "law of nature" mathematical equations. I assume that the 100 years of work by thousands of scientists has not been in vain: the type of equations that they have come up with DO represent fundamental reality and not just a human viewpoint, although the precise details of "random" quantum events cannot be represented symbolically by mathematical equations.

Re "Your example: E = mc2. This equation means: the invariant mass of a system is the total energy divided by the speed of light squared" (Henk, 2 March, 17:01):

Henk, I think you seem to have closed your mind to the possibility that the symbols in law of nature equations could be interpreted in any other way. I'm suggesting that such equations do not live inside a black box, immune from further investigation. I'm suggesting that what we have are:

1. Distinct symbols representing acknowledged aspects of the underlying reality (e.g. E and m, representing energy and mass information respectively)

2. Strings of symbols representing some sort of relationship or structuring of these acknowledged aspects of the underlying reality (e.g. E = mc2 or E2 = (pc)2 + (mc2)2 )

3. Distinct symbols representing this structuring or relationship, where it is clear that "=" represents a different type of thing to the symbols "+", "-", "ﾃｷ" and "ﾃ--".

Take these "structuring" symbols away from law of nature equations and you are left with nothing: no interconnection or relationship, no "balance" between one category of information and other categories of information. I contend that that the symbols "+ - ﾃｷ ﾃ--" and "=" represent vitally important aspects of the underlying reality that really exist, and should not be ignored or denied.

Rick and qsa, my contention is that no matter what type of mathematics, algebra or geometry you want to use to represent fundamental reality, and no matter what particular mathematical equations you use, the above issue is essentially the same.

Lorraine

* I mean information in appropriately integrated physical systems like particles, plant cells and people; but I don't include computers because:

1. Computers don't have the internal physical interconnections to allow subjective experience (subjective experience of a physical object being summary information about the surrounding reality and the internal reality of the physical object)

2. Computers only deal with REPRESENTED information not information per se.

Hi Lorraine,

yes you make some interesting points. IMHO There aren't just static relations as is the case with a block time universe but a changing universe that is continually recycling itself, which makes the mathematical operators seem very important and relevant, as well as the equals which separates inputs and operators from the output.

When applied to physics the equals sign is (at least sometimes) misleading, as the apparent output may not be the result of inputs and operators alone but also local environment,including historical influences and the behaviour of an observer. So it is more of a leads to something but it is not equivalent to the inputs and the obvious operation upon them alone. That's not saying that there is something more than mathematics happening but the mathematics that applies in nature is completely joined up, as is the material universe.Taking a look at an isolated system is not the same as what is going on entirely.

In mathematics it doesn't matter how the number of things are organised 5 separate carbon atoms are equal to 5 carbon atoms in a ring. As 5=5, but in chemistry the organisation of the atoms makes a big difference to how the chemical behaves and they certainly are not equal or the same. The role of organisation as information is very important in nature, It is very obvious in biology, especially morphogenesis. It has been rather neglected by physics.

Sorry, that previous post by Anonymous is mine.

Lorraine,

Twice now, you have explained in detail your question. None of my reply (or others') seems to help you in your research. Yet, we keep trying.

Your question I believe is; Since these mathematical symbols are part of highly successful equations describing the behavior of the universe, why would they not also be part of the way the universe operates? The question then becomes; Does the universe effect an operation similar to (an example) the addition?

To this question about the addition, I have offered a reply in the following. A planet will gain mass by acquiring matter, not by computing the available mass in the vicinity. The physical addition for the universe consists in bringing matter together and therefore it has a dimensional aspect to it. The added matter also add in collocation its properties to that of the whole planet; the total mass and gravitational force of the planet are changed. Of course, it does not tell you why the planet acquires material from the vicinity, i.e. the root cause of gravitation.

In conclusion, the universe may effect some of these mathematical operations, but, they remain just on a need to know basis for an intellect. The main question is a "why" calling for a logical operation (spontaneous as the universe is) on elements of a same dynamical nature. This you may do by stepping into the black box by understanding and acknowledging its unavoidable rules and content.

p.s. The appeal of a question often reside in the prospect of a specific answer. Instead of letting us all pull ourselves slowly to this prospected answer .... why don't you let us know what YOU think?

Cheers,

Marcel,

Hi Marcel and Georgina,

Re "Does the universe effect an operation similar to (an example) the addition? ":

When it comes to the calculation seemingly implied by a "" in law of nature mathematical equations, it seems clear that the underlying reality does not need any space, time or energy to perform the calculation - the underlying reality is seemingly not even performing calculations at all!

Calculations involve REPRESENTATIONS of reality at all stages of the process, so when physicists perform calculations involving law of nature mathematical equations, the result is a symbolic representation of a physical outcome. This is in contrast to actual reality, where physical outcomes are the genuine article, and seemingly no calculations take place.

Seemingly, the underlying reality only requires the "structure" represented by law of nature equations to be in place (together with preliminary conditions) in order to produce a physical outcome. So while a "" looks like a mathematical operator to us humans, I would suggest that it actually represents a structuring relationship in the underlying reality.

Lorraine

Nice thread,

I have been researching and writing along these lines for the last four years. The fruit of that labour is that I have effectively turned the MUH into a full fledged scientific theory, called the Theory of the Structural Multiverse. The conceptual coherence and empirical tractability both gain significantly as a result. To my knowledge, I give the most complete discussion of the evidence yet available. I am looking for peer reviews at this point. If anyone is interested in reading the proposal or a few chapters and give comments please let me know.

Thanks,

Colin

All,

Could the question be framed as follows?

"Are there 'physical laws' which are not and cannot be 'mathematical identities'?"

If so, why should Nature follow our mathematical calculations and derivations? I have argued in The Thermodynamics in Planck's Law and in my fqxi essay The Metaphysics of Physics. that all 'physical laws' can and should be derived as 'mathematical identities'. I have shown, for example, Planck's Law for blackbody radiation is a 'mathematical truism' and not a 'physical law' per se depending on the physical assumption of 'energy quanta'. This explains why the experimental spectrum is so identical to the theoretical curve.

Constantinos