> On 17 Nov 2018, at 03:58, John Clark <johnkcl...@gmail.com> wrote:
> On Fri, Nov 16, 2018 at 12:42 PM Bruno Marchal <marc...@ulb.ac.be 
> <mailto:marc...@ulb.ac.be>> wrote:
> > A practical difficulty here is that logicians used the term model like 
> > painters: the model is the reality
> Mathematician can use one part of mathematics to model another part, for 
> example Descartes found a way for geometry to model algebra, and those 2 
> things can have equal complexity;

Those are representations, which is related notion,  but it is different from 
the notion of model in logic. The notion of model “modelises” the notion of 
reality. A “concrete group” is a model of the finite syntactical theory of 
group. The structure (N, 0, +, *) is a model, among infinitely many others or 
the Peano Syntactical theory of numbers.

> but that like using English to talk about the English word "cat". Whenever 
> mathematics tries to model something that is not itself, like something 
> physical,

Which might be part of mathematics. Unless you assume a physical reality out of 
mathematics, or out of the mind of the Turing machine, which “live” in the 
standard model of arithmetic, in fact in all models of arithmetic.

> it always comes off looking second best because mathematics is just a 
> language, a very very good language for describing physical law but a 
> language nevertheless.   

That is not correct. We do use a language, but the reality (model(s)) are not a 
language. In logic, we have to distinguish the language (which decide which 
sentences are grammatically correct formula) from a theory, which is a finite 
(or recursively enumerable) set of formula (called axioms), and a model, or 
reality, which is a mathematical structure, or something else, which satisfies 
the axioms, and such that the inference rule preserves that satisfaction 

> But, I hear you say, the numbers 11 and 13 are prime and that fact is 
> unchanging and eternal!  Well yes, but the English words "cat" and "bat" 
> rhyme and that fact is also unchanging and eternal.

Not in the same sense, and if you make things precise, for mechanism, a theory 
with bat and cat rhyming can be Turing universal, and then it is just a change 
of basic ontology. The idea here is to use a theory where everyone agree on the 
simple operational meaning. See the combinator theory thread for a different 
example than arithmetic. 

> > I alluded to the fact that you can identify (by clear definable bijection) 
> > a model with the set of (Gödel number) of all true sentences in (the 
> > standard model of) arithmetic.
> Mathematics can't even identify all true sentences about arithmetic much less 
> become the master of physical reality. We know  the sentence "the 4th Busy 
> Beaver number is 107" belongs in the set of true sentences, but what about 
> "the 5th Busy Beaver number is 47,176,870"?  It's either true or its not but 
> will you or I anybody or anything ever know which one?  Nobody knows and 
> nobody knows if we'll ever know, but we do know that nothing will ever know 
> what the 8000th Busy Beaver number is even though its well defined and finite.

You make my point. The value of the busy beaver function is arithmetical well 
defined, but not computable, which illustrates that the arithmetical reality 
kicks back, and is indeed very huge. After Gödel we know that we can only 
scratch that kind of reality. Yet, its conceptually clarity make us accepting 
realism, which is basically the idea that the excluded principle is valid there.

> >You already need 2+2=4 to make sense of matter,
> Recent studies see to indicate that without a working brain a person's IQ 
> tends to drop rather dramatically, so you've got it precisely backwards yet 
> again,  you need matter to make sense of 2+2=4 or to male sense of anything 
> at all.

Assuming Aristotle theology (materialism), but there is no evidence, and it is 
refuted by Mechanism.

The IQ test can only observe the 3-1 person, not the 1-1 person. The 1p can 
only associate its own consciousness to an infinity of representation of its 
body in arithmetic, and the notion of “having no brain” is relative to the 
computations, so your argument needs your ontological commitment in some 
primary matter, for which there is no evidence found yet. 

> > But you don’t need silicon,
> True, carbon and carbon compounds will also work. 
> > or “being made-of” to define the numbers.
> You need a brain made of some sort of matter to define numbers

Sure. But the numbers does not need me to exist. 2+2=4 even if I was not born.

You seem to confuse “I can define number” and “the number itself”. Indeed a 
brain to grasp 2+2=4, but I need a brain to observe a far away galaxy. Yet we 
agree that the galaxies do not need Hubble to exist, and it is the same with 
the numbers, given that we will explain the brain by the notion of digital 
machine, and I have to assume the numbers at the start to make sense of the 
terms like machine, brain, etc.

You are only keeping Mouloud your personal materialist credo, but that is not 
how to proceed when doing science.

> or to define anything at all, not that there is anything special or even very 
> interesting in the act of definition, you need a brain made of matter to do 
> anything. 
> > If 2+2=4 depends on matter, tell me how a magnetic field, or a 
> > electromagnetic field, or a gravitational field, or any physical field 
> > could pertubate 2+2=4.
> 2+2=4 is a description in the language of mathematics about how some physical 
> properties behave. For example, the mass of 2 protons and the mass 2 more 
> protons equals the mass of 4 protons. But 2+2=4 doesn't work for everything, 
> the temperature of 2 hot water bottles and 2 hot water bottles does not equal 
> the temperature of 4  hot water bottles. Temperature doesn't add up in the 
> same way that mass does, a different description is needed to describe what's 
> going on.

No problem. 2+2=4 should not be applied in all context, of course. 

> >computations, can be defined in [blah blah]
> Who cares??  Definitions are just a human convention, a definition of a 
> computation can't compute and a definition of a airliner can fly you to 
> London.

A definition of a computation is not a computation. But can be used to show 
that all computation are done in the models of arithmetic. 

> > You confuse the [blah blah]
> No, you confuse the difference between a cat and the word "cat" . The 
> difference is one can have kittens but a word can’t.

Yes, that is what I insist. So please stop confusing the language “2+2=4” with 
the fact that 2+2=4. Same for the computations. Arithmetic contains all 
description of computations should not be confused with the fact that all 
computations are also realised, done, executed, through the truth of the number 
relations. “Cat” is not a cat, “one” is not the number 1 either.

> > the models/realities intended will be usually much more complex, as you 
> > said above.
> Mathematical models are ALWAYS simpler and less rich than the physical 
> reality they try to represent.

With “model” used in the sense of the physicist. That is true for arithmetic 
too, as you allude above. The arithmetical reality (model) is far more complex 
than any theories of arithmetic.

> So why in the world would you say the physics is modeling the mathematics 
> when its obvious that the mathematics is trying, with limited success, to 
> model the physics?    

No one says that physics model mathematics. With mechanism, physics is 
reducible to the theology of numbers, which is not reducible to any other 
theory, and is of course not an axiomatisable theory, except, miraculously for 
its propositional parts, as we know since Solovay 1976.


> John K Clark
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