The limits of mathematics

This isn't about pure math as such, but addresses the issue of whether math can fully model the physical world. I say, no. For example, note that due to its being a matter of logical necessity, math cannot produce true randomness of the sort many consider manifested by quantum behavior. What I mean by "deterministic" math is that the math process can't actually *produce* the random results. Just saying "this random variable has no specific value" etc. is "cheating" (in the sense philosophers use it), because you have to "put in the values by hand." Such math either produces "results" which are the probability distributions - not actual sequences of results - or in actual application, the user "cheats" by using some outside source of randomness or pseudo-randomness like digits of roots. (Such sequences are themselves of course, wholly determined by the process - they just have the right mix that is not predictable to anyone not knowing what they came from. In that sense, they merely appear "random.") I think most philosophers of the foundations of mathematics would agree with me. As for MWI as an doge, I still ask: why doesn't the initial beam splitter of a MZI split the wave into two worlds, thus preventing the later interference that we find?

Dear Ian,

Yes I know that distance and time are relative if we bring Relativity Theory into play. That is not a problem to me. I won't be able to say anything more than that relativity type problems and analogous forms of their respective equations can be drived without Relativity Theory. The only indication that I can put forward in a simple way is to suggest that transform equations are not safe mathematics. They can be made to fit whatever the theorists makes them fit. If the Theory of Relativity was correct, then I should think that it could be derived directly from the fundamentals without the use of transforms. That is what I suggest is both required and possible to do.

Probably something that is of more immediate interest is the fact that you mentioned electric charge. I had it in my message and then took it out. Since you astutely brought it into play, then I will say that it does receive a fate analogous to that which mass suffered. It also does not deserve indefinable status. In support of this statement, I will suggest that physics equations, from an empirical viewpoint, never include causes. Electric charge is a theoretical cause. The possibility of it being artificial is raised by the fact that it was theoretically identified as what were otherwise unknown quantities that appear in Coulomb's equation. This is not easy to say; but it was another theoretical guess.

Causes are not a part of physics equations except in the sense that they are all represented by the equal signs. I guess this is probably enough said. I have suggested before that almost everything could change. If it doesn't look right to you, I understand. It might be better to give it a rest for a while. I do appreciate your patience and thoughtfulness.

James

Can I get a scholar to do the arithmatic.

?

to get the equation for the big bang we look at a Godel universe where time is a contradiction.

And we look at the penrose equation for a black hole in a Godel universe.

And reverse it to get a non contradictory equation for a big bang in our universe.

That is without the meanningless infnities.

Hi Ken,

Why do you deny some obvious restrictions in reality, which got lost in mathematics? In particular I refer to extensions of natural numbers.

Hopefully you will agree that there are no negative or imaginary tangible items. You might have debts but definitely no negative coins in your pocket.

For some implications you might read or reread at least my essays 369 and 527.

Eckard

Dear James,

Yesterday on Feb.21 you wrote to Ian: "The only indication that I can put forward in a simple way is to suggest that transform equations are not safe mathematics."

Maybe, you did not even take into consideration the absolutely safe practice of engineers who know what they are doing. They first transform physical quantities from a real domain into an artificial, in particular a complex domain and in the end they safely return into the real domain by means of an inverse transform.

Perhaps you meant that transforms itself are safe, provided they are correctly performed, while interpretation in the artificial domain are guesswork and sometimes prone to be fallacious.

Regards,

Eckard

Dear Eckard Blumschein,

"...Perhaps you meant that transforms itself are safe, provided they are correctly performed, while interpretation in the artificial domain are guesswork and sometimes prone to be fallacious. ..."

You are of course correct. Thank you for this clarification.

James

Dear Ian,

"...Actually, special relativity can be derived without reference to the transforms. Not to beat a dead horse, but Moore does it graphically in his text. .."

I haven't seen all the sources there are to learn from; however, I would suspect that he probably went to advanced theory or end points of theory, conservation type properties, and worked his way back from them. If I am wrong than I apologize for misrepresenting what he has done. I was prompted to say this because every explanation I have seen of Relativity Theory, other than its original type derivation, usually begins with theoretically advanced properties that owe their origin to the assumed adoption of Relativity Theory. It is a practice of beginning at the end and working back toward some set of 'fundamental properties' as established by theory. .

"...Can you clarify one thing for me that seems to be hindering my full understanding of what you're saying: when you say equations in physics don't have "causes" what do you mean exactly by "cause" in this case? If it means what I think it means, I would say that most of these "causeless" equations are empirically derived, i.e. they are the way they are because that's what experiment seems to indicate. Maybe this is the point you are trying to get at but I'm not sure. ..."

Thank you for this question. I spent the weekend thinking over what I had said. It seemed clear to me; however, when I thought in terms of explaining it to someone else in detail, I had to think about it a lot. I am still thinking about it and now also writing about it. We do not know what cause is and yet it must be indicated by empirical properties, not theoretical ones, somewhere in the equations. I have already been challenging myself to support what I said. When I can say something more clear about it, or perhaps modify or rescind it, then I will respond.

I also thought that I was once again veering off into my own ideas about possible solutions. The point I really wanted to make, in answer to your original question, was that I think mathematics leaves reality behind when the theorist begins to guess about new extra givens and interjects them into the original equations by identifying unverifiable properties and even more than this, interjecting indefinable units into the otherwise empirically clean equations.

You certainly are patient with me. I acknowledge that I know less than do you about complex theoretical physics and the fullness of empirical knowledge. I do think something is fundamentally wrong; however, I know and appreciate the necessity for demonstrating it and having it challenged. Thank you for your time.

James

Ian,

Mathematics is not fundamental because they are born out of logic. Logic is more fundamental to the universe. The universe only requires logic in order to work. But the unnatural vantage point of the observer creates and requires numbers greater than one (1).

Mathematics is the extension of logic for the conscious mind. Sure, nice geometric and mathematical structures emerges in nature from the effect of logic on large numbers. But again, the appreciation of those same structures is but for the conscious mind.

Marcel,

Dear Ian,

"Basically it comes from three things: a) the principle of relativity which simply says that the laws of physics ought to look the same in all inertial reference frames, b) recognition that time is not absolute like Newton thought, and c) assuming the speed of light is a maximum."

Ok, that doesn't sound like higher level theory. A and C are fine. Why is B not an example of using Relativity theory to support Relativity Theory?

James

MI,

You wrote: "If mathematics is discovered, this implies that the objects in question preexisted and existed before anyone was there to have thought about them. Believing such a thing seems like an act of faith (or perhaps confusing the model with reality)."

Non sequitur.

Georg Cantor made a celebrated statement: "The essence is just its freedom." He did not manage escaping the small city Halle where he died in a madhouse, not by means of having got the CH directly from god, and not even by means of this populism.

I vote for the opposite: Mathematics must avoid arbitrary fabrications. Meanwhile the seems to be a lot of arbitrarily founded guesswork in mathematics and even more in its applications on physics. Let me mention a bit of very obvious nonsense:

Rotate or blow up a point (Confusing the point with a phasor or a tiny sphere)

Singular treatment of a real number, e.g. |sign(0)| nonsensically = 0.

Denial of restriction for physical quantities to positive real values

Spacetime is thought to from minus infinity to plus infinity, amen.

Heaviside was cheeky enough as to call mathematics an empirical science. Indeed, it arose from applied reasoning. Many animals are already intelligent enough as to learn by trial and error repetitious patterns. Consequent further steps were the abilities to count and to trust in causal explanation. I see it neither justified to deny laws of nature nor to follow Quine/Putnam.

Evidently the relationship between e, pi, and i was discovered step by step. However, you must not infer that objects exist forward and backward "in time".

Eckard

Ian,

One reason for me to deal in depth with the history of introduction and interpretation of negative, imaginary and complex numbers, evanescent modes, spacetime, apparent power and the like were striking apparent symmetries and those "experts" who offered physical interpretations to mere mathematical constructs instead of admitting redundancies that imply arbitrary choices.

I found out that the mathematicians strove for an as general as possible point of view and therefore neglected R as an unimportant special case while virtually all physical items and quantities are basically restricted to positive values except for logarithmic scale or use of an arbitrarily chosen shift of the origin. Isn't this a serious and risky deviation?

You wrote: "it is entirely possible to do special relativity solely with the graphical method (which actually is not unique to him) and never know the transformations at all." Well, likewise one can graphically represent the orthogonality between the voltage at an inductor and the current through it. Here I agree with James Putnam in that the graphical relationship is ambiguous on whether the voltage causes the current or vice versa. My understanding of reality tells me that the primary relationships are not differential equations but integrations and therefore in this case the voltage or in case of motion the force are the action while current or velocity, respectively, are reactions.

In other words, the graphic method just hides the obligation to correctly perform an inverse transform into the domain of reality. As shown by Heisenberg, matrix representation may also obscure the essentials.

Ironically, all fathers of quantum mechanics went wrong just because they cared for physical correctness. They argued that frequency must be a positive and real quantity. However, exactly this good intention led to an apparent symmetry of wave-function, a wrong interpretation that contradicts all experience.

Eckard

Ian,

Maybe "consciousness" is a bit too much to ask. Lets just say an integrated geometric point of view. A camera with proper optics may capture a point of view e.g. the honeycomb structure in a picture. Does it appreciate it ??? Don't think so. Why "integrated" ? Because, if we could discriminate every single incoming photon, all we would see is a bunch of scintillations... Like the film emulsion or the CCD chip, we must accumulate the incoming data in order to be able to form an image. We have to remember also that the capturing of photons to make a picture is not related to the actual distance to the origin of the photon. Otherwise, all we'd see is a slice of the landscape at some distance. In other words, we can see at a glance in the same moment of perception both the Sun at 8 light minutes from us and the Moon at about half a light second away. These are the specs we use to view the universe.

So, either you are one (1) minding your own business, or you are a spectator with a specific point of view that allows you to count numbers higher than one. Yes, it is kind of a metaphysical reasoning: You are, or you watch what is. I can see four levels here. 1) you are one minding your business. 2) you are a spectator that has a 2D point of view 3) your mind allows you to figure out a 3D world out of your 2D point of view. 4) you forget about the D's because you know the universe is not a point of view and you understand it as 1) does.

Marcel,

I know that you stated that you wanted to approach the subject from a non-speculative standpoint but some of the things you are querying are dependant on the Ontologial status one places on Mathermatics.

Is Mathematics something that is discovered or created? Is mathematics used as an inferential model of reality or a description of it?

What is the mathematical model under scrutiny meant to describe? What is the aim and scope? As an example, the foundation of much of the mathematics employed in modern theoretical physics is based on the implcit and explcit use of the imaginary number system and it's subsequent analysis. There is no physical or real correlation with imaginary numbers and phenomenon or measureable quanitities in the real world, but the theoretical use of the imaginary number system is critical to the mathematical formalism present in much of Physics.

Also, historically, when singularities or divergences are encountered in mathematical physics, they are simply ignored or brushed aside. A classic case is renormalization in QED and field theories.