On 9/27/2018 9:29 PM, Bruce Kellett wrote:
From: *Bruno Marchal* <marc...@ulb.ac.be <mailto:marc...@ulb.ac.be>>
But now, let us move forward. Stop saying “realism or platonism”, in
pour metaphysical context this lead to misunderstanding.
Assuming classical arithmetic = arithmetical realism.
It is becoming clear that we have very different understandings of
what is meant by arithmetical (or mathematical) realism. I gathered a
few statements about realism from your recent post -- included here:
"Realism = classical. Realism means that I use the axiom (A v ~A)."
"All scientific theories use arithmetical realism, but you are still
using it in a philosophical/metaphysical sense, when it means simply
that we accept the excluded middle principle in arithmetic."
"Of course, given what I mean by arithmetical realism (which I thought
I already told you), this would mean that you reject the use of (A v
~A) in arithmetic"
My understanding of 'realism' comes from the idea of scientific
realism. This can have a number of nuanced interpretations, but the
basic idea of scientific realism is a cluster of views about the
nature of scientific theories and theorizing. A common core might be
the following:
(1) The aim of scientific inquiry is to produce theories that provide
description of the world that are literally true.
(2) Theories in the 'mature sciences' are usually approximately true,
and the entities postulated by those theories usually exist.
One of the most popular arguments for scientific realism is the
so-called 'miracle argument'. Following Putnam, scientific realism is
capable of explaining why a predictively successful theory is
predictively successful, whereas the success of a theory would be
miraculous if scientific realism were not true.
Stathis Psillos adds a metaphysical component: the world has a
definite mind-independent structure; and a semantic component:
scientific theories are truth-conditioned descriptions of their
intended domain, so the theoretical terms in theories have factual
reference -- the unobservable entities they posit populate the world
-- form the 'furniture' of reality.
The Oxford Dictionary of Philosophy, in the section on the philosophy
of mathematics, gives the following definitions:
"There are two distinct types of realism in the philosophy of
mathematics. Realism-in-ontology is the view that the subject matter
of mathematics is the realm of objects that exist independent of the
mind, conventions, and language of the mathematician. Most advocates
of this view hold that mathematical objects -- numbers, functions,
points, sets, etc. -- are abstract, eternal, and do not enter into
causal relationships with material objects. Because of this,
realism-in-ontology is sometimes called platonism.
"Realism-in-truth-value is the view that unambiguous assertions of
mathematics are non-vacuously true or false, independent of the mind,
language, and conventions of the mathematician. (This would seem to be
close to the view that you, Bruno, espouse.)
"There is a natural connection between the two varieties of realism.
Consider the following statement:
'There is a prime number greater than 1,000,000.'
"The realist-in-truth-value holds that this is an objective truth. But
what does it mean? Prima facie, '1,000,000' is a singular terms, and
'prime number' is a common noun. If the surface grammar of this
sentence reflects its logical form, and if 'there is' means 'there
exists', then the sentence entails that both the number 1,000,000 and
a greater prime number exist. For the realist-in-truth-value, this
existence is objective, and so we are led to realism-in-ontology. In
sum, if one is a realist-in-truth-value, then realism-in-ontology is
the result of taking mathematical assertions at face value."
Other references that I have looked up, such as entries in the
Stanford Encyclopedia of Philosophy on "Realism" and "Platonism in the
Philosophy of Mathematics", say similar things. Though, of course,
there are probably more nuances in the understanding of mathematical
realism than there are philosophers of mathematics.
Given the above references, I think it should be clear why I say
"realism or platonism", and refer to "an independently existing
mathematical realm". In Western philosophy at least, that is what
realism in mathematics means -- although things might be different in
Gallic philosophy.
It seems that your idea of arithmetical (mathematical) realism is
entirely from classical logic and is, therefore, essentially a
'realism-in-truth-value' understanding. It is interesting, in that
case, that you make no reference to mathematical objects. You claim
that the truth of propositions such as '2+2=4' is independent of the
mind, language, and conventions of arithmetic, as in the definition of
'realist-in-truth-value' above. But you do not seem to go the
additional step of saying that mathematical objects, numbers and so
on, are objects that actually exist (which would be a form of
platonism). If you want to reject platonism, and the idea that
mathematical concepts are objects that actually exist -- that there is
a mathematical realm of objects that exist independently of any
physical existence -- then I suppose you are entitled to any view that
you wish to hold. But you cannot claim that any such view is uniquely
necessary.
If you reject platonism, it is hard to see how you can make sense of
claims such as "All calculations exist in arithmetic", or that physics
arises from the statistics of computations in the universal
dovetailer. Since I reject all forms of arithmetical realism,
particularly platonism, I do not think that your arguments for 'comp'
have any merit.
However, the philosophy of mathematics is not an area in which I have
had any particular interest, so apart from rejecting mathematical
realism and platonism, I do not have any strong views about which of
the many alternatives on offer might be an acceptable philosophical
attitude to arithmetic.
Bruce
I don't see any problem in saying that "2+2=4" while denying that
numbers exist. ISTM analogous to saying "A unicorn has a single horn."
without implying that unicorns exist. The truth that is maintained by
mathematical proof is just a marker which is analytically preserved by
the rules of inference. The fact that there are true but unprovable
sentences in arithmetic, in just a logical inference in
meta-mathematics. It's not something outside mathematics that is true
in the sense that ice is cold.
Brent
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