On Monday, November 19, 2018 at 5:24:48 AM UTC-6, Bruno Marchal wrote:
>
>
> On 17 Nov 2018, at 03:58, John Clark <johnk...@gmail.com <javascript:>> 
> wrote:
>
> On Fri, Nov 16, 2018 at 12:42 PM Bruno Marchal <mar...@ulb.ac.be 
> <javascript:>> 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 relation.
>
>
>
>
>
> 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.
>
> Bruno
>
> John K Clark
>
>
 

There are two types of people:

Those who say "There are mathematical objects."
Those who say "There are no mathematical objects."

I was just thinking about a certain irony about Hartry Field*: His last 
name has meanings in both physics and mathematics, and he is the founder o*f 
nominalized semantics* for theories of physics — where a 'field' may be 
nominalized away!

* https://en.wikipedia.org/wiki/Hartry_Field
  
 
https://books.google.com/books/about/Science_Without_Numbers.html?id=Exc1DQAAQBAJ

- pt

 

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