The limits of mathematics

Ian,

you solicited thoughts on " ... the Quine-Putnam indispensability argument which basically says that if we believe in the concreteness of the physical theories described by mathematical objects then we also ought to believe in the concreteness of those mathematical objects themselves."

"Concrete" as opposed to what? Abstract? Then is there such a thing as "concrete" language? That is, do symbols stand for themselves only, or are they independent of the objects for which they stand?

My view: language is independent of meaning. It follows therefore that physical (mathematical) theories are no more concrete than the language (mathematics) that describes them. There is a distinction to be made between objects and meaning; there is no such distinction between a (physical) theory that maps symbols to symbols and a (mathematical) theory that maps symbols to objects. In other words, physical objects and mathematical objects are both symbolic representations independent of phenomenological observation. Thus, physical science is necessarily an open system, progressing toward what Popper called verisimilitude, an asymptotic approach to truth, and never offering a completely closed judgment of truth, a proof of its conclusions. Mathematical science necessarily offers closed judgments based on axiomatic deduction, and theorems (true mathematical statements) are proven in the domains to which they apply. The physical domain--being the whole observed, and even perhaps the unobserved, universe--may not be subject to such axiomatization. How would we know in any case?--Goedel taught us that no set of axioms is sufficient to prove its own self-consistency; there always exist true statements that cannot be deduced from the axioms.

That being said, there are serious attempts to recast mathematics as an experimental science. Chaitin, Wolfram, et al, may really lead us to a common point of closure between what we say about the world and what the world says back.

Tom

Dear Ian,

This is what I think: We only know about effects because that is what empirical evidence consists of. We do not know what cause is. It is our lack of understanding about the nature of cause that makes theories necessary. The theorist imagines what causes they think could be responsible for the effects they see. I say this because photons causes changes of velocity. We do not know why they do this, so, we imagine that they have properties that we interpret to be causes.

When I speak about our learning about the operation of the universe by observing patterns in changes of velocity, I mean that that is all we ever directly observe. I avoid inventing indefinable properties. I also avoid inventing causes. Names of convenience are fine to use. What is not helpful is to force our theoretical guesses about causes into empirically determined equations through the means of indefinable invented units.

There is no escaping the necessity for one unknowable fundamental cause. Saying that it is unknowable is not meant to suggest that we cannot identify it and name it. Unknowable is only intended to acknowledge that it is a given without explanation of its origin or ability to cause action. One miracle such as this is enough. Anymore givens are just taking us further away from recognizing that fundamental unity exists right from the start of the universe and this should be reflected in theoretical physics. Fundamental unity should be a part of physics theory right from its start.

I am attempting to make just two points at this time. One is that all empirical information is gathered in the form of patterns of changes of velocity of objects. The second is that I free mass from its artificial theoretical definition, I do not do away with it. It remains as the m in f=ma. All that has changed is that that simple equation has been returned to its empirical roots. The changes that follow because of this act affect almost all of physics theory.

I will not use my own theoretical work to demonstrate this by bringing those ideas here. This forum is not the proper place for me to expound on a new theory. So, I will instead see if I can write a few messages that stick to this idea of returning all physics equations to their empirical roots. I can touch on ideas such as energy and momentum.

James

Ian,

You wrote, "So how much of what underlies mathematics is 'real?' Since much of mathematics can be described as a process, are these underlying processes real?"

In those terms, I think we _have_ to take mathematics as an experimental science. That is, as you implied elsewhere in your post, there is no reality in arbitrarily chosen symbols and their manipulation; as an art, mathematics is not "about" anything, any more than natural language--that the syntax of a statement may be entirely correct, does not imply meaning.

If we take meaning as real (that is what I mean by 'what the world says back to us'), then the meaning that comes from a computer programming language is as real as the substrate on which the program runs, because that is the process, the mechanical throughput, that mimics all physical processes. I.e., information flows continuously over the substrate, though the exchange of information between nodes is in discrete units. The bad news is that Chaitin has discovered an algorithmically defined but uncomputable number (Omega) whose digital expansion is normal but whose value depends on the computer language running the algorithm. So even our arithmetic, in this context, possesses a degree of built in uncertainty. This raises the question of whether a complete one-to-one relation between language and meaning is even possible.

Tom

Dear Ian,

That is a undergraduate textbook. I do not have it; however, I have formally studied physics and have many books. I was just trying to introduce a different way of deriving theoretical physics. It does not do away with any conservation laws. Theory must adhere to empirical evidence. What it does do is remove all of the, in my opinion, artificial theoretical guesses that are preventing us from achieving unity without having to add on more invented artificial theoretical guesses. Unity should not be an after thought that must be forced onto theory by introducing additional unverifiable properties. Anyway, I have thought a great deal about this and have completed a great deal of work demonstrating it. I think you have been extremely polite to me. I appreciate your interest. Maybe this isn't something to pursue here. It definitely does not embrace the approach to physics the that book teaches.

James

Dear Ian,

No I am not affended. You are as nice a person as I have met here. It just that the question had to do with discovering when math no longer is representing reality. I have studied and used theoretical physics. I am not a physicist, but I think there is something important to be pointed out about what physics theory really represents. The difficulty to overcome is that most theory is believed to be reality. I was suggesting that none of theory represents reality. When I mentioned that the definition of mass could be chosen differently, I think it should be clear that the original choice made was a guess. No one could possibly know that mass deserves to be an indefinable property. When I mention that we learn everything via photons and that their information is limited to observing patterns in changes of velocity, I don't think that that is a theoretical statement. Photons begin with changes of velocity and end by causing changes of velocity. We can theorize about what photons hold or contain that allows them to cause changes of velocity; but, we cannot know that. It is theoretical.

My point from beginning to end is difficult to make clear against the common belief in theory. This is my point: The best 'theory' is the one that removes all theory. Theory helps us to keep our thoughts straight, but it does the opposite in terms of learning about reality. I do not mean that it is not useful. I only mean that it is invented as a means to proceed with analyzing physics knowledge without needing to understand the nature of cause. We do not know what cause is. Everything that is attributed as being a cause is invented. This practice of theorizing may be useful, but in terms of understanding the nature of the universe, it is misleading and at times misguided. If we wish to allow math to truly serve us in learning about the nature of the universe, then we must let it take its own course and not steer it this way and that way by interjecting theoretical ideas.

The difficulty with suggesting this kind of approach is that theory clearly stands in the way. It is believed to be true. How can one say that everything could be changed when it is believed that everything is known to be true? Anyway I work for the change. That is why I work alone. That is why I put my work on the Internet. There is no other way to let it out. There are no other books or sources to point to. I am not insisting that others should quickly recognize that I am correct. I may not be correct. However, I think it should be possible to challenge theory without having some theory put forward as evidence of the correctness of other theory. I think that theory can never prove theory. The mathematics will work out properly, but the mathematics has long since become the tool of the theorist and the servant of theory.

James

James

Ian,

I did not get the impression you are hating yourself as admitted Teller Ede.

Names can indeed be misleading. For instance, James Putnam's attitude seems to be more appealing at least to me if I compare him with Hilary Putnam who is said to have a reputation for frequently changing his position. Quine is also suspect to me for some reasons. In particular he justified some transfinite set theory that does definitely not have any bearing in physics. Isn't blind trust in "our (currently) best scientific theories" rather subservient?

In your essay PEP seems to play a role. I am not sure whether PEPing up quantum mechanics will resolve many problems. When I wrote an unpublished manuscript "A still valid argument by Ritz", I offered a guess that explains the PEP in a quite simple manner. Somewhere in the discussion to my essays I provided a link.

Regards,

Eckard

Dear Ian,

It is difficult for any of us to say those things that we think are important in a compact form. I will give it a try:

I think that the point at which math starts to diverge is when we introduce indefinable units. However, let me say this in a different way. Returning to f=ma (I purposefully do not begin with f=dP/dT), the only part of this equation that can be explained is acceleration. That is because it consists of measurements of distance and time. The rest of the equation is unexplained. Force is the unexplained cause for acceleration. Mass is the unexplained cause for the variations of that acceleration. Symbols and names of convenience are used to to form this equation. This practice is helpful without causing problems.

The properties involved are all empirical properties. Two of them are distance and time. These are real directly known indefinable properties. The other two properties are force and resistance to force. These two are indirect empirical properties. Both force and resistance to force have unobservable natures. We know from empirical evidence that they exist; however, all of that data consists of measurements of distance and time. Measurements of distance and time are the extent of direct empirical evidence.

All higher level theory will consist of combinations of these four properties. Everything else added on changing this simple perspective is theoretical. The distortional effects of theory are first introduced by chosing to define mass as an indefinable property requiring its own indefinable units. From this point on all theory using this definition of mass is no more correct than is that first definition. It is important then to get the definition of mass correct right from the start.

I have described mass as the cause of resistance to force. We do not know the nature of any cause. Cause is hidden from us except through its effects. I think that the best approach to reconsidering the definition of mass is to add nothing on to the equation and rely only upon our use of empirical evidence in the form of distance and time and their units only. That is the concept that I used in my earlier message where I showed one possible new way to interpret mass. I have gone a long way by trying out this approach. Perhaps I may have gone astray myself; however, what I can say for certain is that: This idea does not quickly fall apart, it just keeps on going and going.

I will stop and see what you think.

James

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