> On 25 Feb 2019, at 11:52, Philip Thrift <cloudver...@gmail.com> wrote:
> 
> 
> 
> On Monday, February 25, 2019 at 3:34:15 AM UTC-6, Bruno Marchal wrote:
> 
>> On 22 Feb 2019, at 18:44, Philip Thrift <cloud...@gmail.com <javascript:>> 
>> wrote:
>> 
>> Some accept the possibility that there can be something that is immaterial.
> 
> Yes. We call them “mathematician”.
> 
> Bruno
> 
> 
> 
> This recent thesis I came across
> 
> Application and ontology in mathematics: a defence [defense] of fictionalism
> http://etheses.whiterose.ac.uk/18636/
> 
> leads to
> 
> pure mathematicians may be immaterialists,
> but applied mathematicians are materialists.
> 
> 
> 
> Abstract
> 
> The aim of this thesis is to defend fictionalism as a response to the 
> mathematical placement problem. As we will see, against the backdrop of 
> philosophical naturalism, it is difficult to see how to fit mathematical 
> objects into our best total scientific theory.
> 
Which “best total scientific theory”? Theology is still taboo, and there is no 
coherent physical theory of the universe, nor clear argument why that would 
exist.

Assuming mechanism, assuming more than the natural numbers, or than the 
combinators, or the programs, is just speculation on undecidable metaphysical 
ontologies. No doubt is put on physics as an art to put relevant order on the 
observable and predictable, but it can’t be the fundamental science. It has to 
be derived from the theology, that is from G* (actually from qZ1*, the 
observable-mode of self-reference, motivated through thought experience and/or 
Theaetetus/Parmenides/Moderatus/Plotinus. (It is an old alternative viewing of 
reality).



> On the other hand, the indispensability argument seems to suggest that 
> science itself mandates ontological commitment to mathematical entities.
> 

Not really. It needs we agree on some basic starting simple relation.

As I have demonstrate recently here, the relation 

Kxy = x
Sxyz = xz(yz)

Are enough. But classical logic + Robinson arithmetic is enough.

No need of ontological commitment other that not denying what we found almost 
obvious in primary school. You can remain formal, but it is simpler to do a bit 
of math and get the intuition that indeed it kick back and, well, 2+2 is not 
equal to 5.


> My goal is to undermine the indispensability argument by presenting an 
> account of applied mathematics as a kind of revolutionary prop-oriented 
> make-believe, the content of which is given by a mapping account of 
> mathematical applications. This kind of fictionalism faces a number of 
> challenges from various quarters. To begin with, we will have to face the 
> challenge of a different kind of indispensability argument, one that draws 
> ontological conclusions from the role of mathematical objects in scientific 
> explanations. We will then examine one recent theory of mathematical 
> scientific representation, and discover that the resulting position is 
> Platonistic. At this point we will introduce our fictionalist account, and 
> see that it defuses the Platonist consequences of mathematical 
> representation. The closing chapters of the thesis then take a 
> metaphilosophical turn. The legitmacy of a fictionalist response to the 
> mathematical placement problem is open to challenge from a metaphilosophical 
> perspective in two different ways: on the one hand, some modern pragmatists 
> have argued that this kind of metaphysics relies on questionable assumptions 
> about how langauge works. On the other, some modern philosophers have 
> developed forms of metaontological anti-realism that they believe undermine 
> the legitimacy of philosophical work in metaphysics. In the final two 
> chapters I defend the fictionalist account developed here against these 
> sceptical claims. I conclude that the fictionalist account of applied 
> mathematics offered here is our best hope for coping with the mathematical 
> placement problem. 
> 
> 


It illustrates the kind of difficulties you can meet when you take for granted 
the idea that the fundamental reality is physical. 

There is only a placement problem for mathematics because people commit 
themselves into some *place* which does not seem to be an hypothesis making 
thing simpler.

Fictionalism does not apply to the arithmetical reality, nor to physics, but to 
the naïve idea of a “physical universe” as being the fundamental reality. The 
theology of the universal machine is a priori quite non Aristotelian: there is 
no Creator, and there is no Creation. Just a universal dreamer which lost 
itself in an infinitely surprising structure and wake up from time to time, or 
from numbers to numbers.

I need no more than a partial applicative algebra, and each choice of the phi_i 
makes N into one, simply by defining an operation “*” in N such that n * m = 
phi_n(m). There exist numbers k and s such that

((k * n) * m) = n
(((s * n) * m) * r) = (n * r) * (m * r),

for all m, n, r in N.

And, the key point, the operation “*” can be defined in the arithmetical 
language, and those statements are, for each n, m, r, provable in RA. I have 
shown that the converse is true. It is a very elegant Turing complete theory. 
With Indexical Digital Mechanism, it is absolutely undecidable if the Universe 
is bigger than the sigma_1 reality. (But here I do a blasphemy: that can only 
be entirely justified by G* *only*!, It is where I have to insist that this is 
presented as a consequence of YD + CT (“yes doctor” + Church-Turing thesis).

Such theories are essentially undecidable. It means that not only they are 
arithmetically incomplete, but all their effective consistent extensions are 
too. They are creative, you cannot capture the semantic in the way it could 
become complete, even in some imaginary domain concevable by the 
machine/theory/number. The universal machine are never entirely satisfied and a 
computation is always an escape forward, but their self-reflection create a 
mess, and illusions.

The sigma_1 arithmetical reality, as seen by the universal numbers which lives 
there, in the first person undetermined sense, is something *very big*. It 
generates infinitely many surprises. There are consistent histories.

Bruno






> 
> 
> - pt
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