[FOM] Richard Epstein's view
Neil Dewar
neil.a.dewar at gmail.com
Sun Mar 18 08:23:28 EDT 2012
I'm not certain (having not read Richard Epstein's book, nor a great deal of Bernays), but it seems to me that this view might have some similarities to comments Bernays makes in "The Philosophy of Mathematics and Hilbert's Programme". For example,
1. Epstein: "I present a view of mathematics as a science like physics or biology, proceeding by abstraction from experience … The theory is applicable in a particular case if what we ignore in abstracting does not matter there."
Bernays: "If we pursue what we mean by the mathematical character of a consideration, it becomes apparent that the typical characteristic is located in a certain mode of abstraction that comes into play. This abstraction, which may be called formal or mathematical abstraction, consists in emphasising and exclusively taking into account the structural elements of an object … that is, the manner of its composition from its constituent parts." (pp238-9)
2. Epstein: "a claim such as 1 + 1 = 2 is not true or false, but only true or false in application. It fails, for example, in the case of one drop of water plus one drop of water = 2 drops of water, so that such an application falls outside the scope of the theory of arithmetic."
Bernays: "For this logical dependence [the logical dependence of the theorems of an axiomatic theory upon the axioms] it does not matter whether or not the axioms placed at the beginning are true statements. The logical dependence represents a purely hypothetical connection: If things are as the axioms claim, then the theorems hold." (p236)
3. Epstein: "On this view numbers are not real but are abstractions from counting and measuring, just as lines in Euclidean geometry are not real but only abstractions from our experience of drawing or sighting lines."
Bernays: "If we want to have … the ordinal numbers as definite objects free from all inessential elements, then … we have to take the mere scheme of the relevant figure of repetition [i.e. a figure of obtaining successors] as an object; this requires a very high abstraction." (p244)
All references are to the translation in Mancosu, "From Brouwer to Hilbert". This isn't intended to suggest that Epstein's view should be taken to belong to the Hilbert programme, but rather that it may coincide with some of the philosophical motivations for that programme.
Cheers
Neil
On 17 Mar 2012, at 17:09, Arnon Avron wrote:
> I would like to repeat an argument already made here in the past
> (like almost any other argument..) concerning views of this sort, no matter
> what "ism" is attached to them: Even people like Epstein
> should accept as meaningful and absolute propositions
> of the sort: "The formal sentence A is/isn't a theorem of
> the formal system T". I do not see how a view of mathematics (or
> science in general) that denies this can be coherent, and in what
> possible sense can the truth of such a proposition be "only true or
> false in application". And of course, once one understands this,
> s/he sees the meaningfulness and absoluteness of
> at least quantifiers-free arithmetics.
>
> Arnon Avron
>
>
> On Fri, Mar 16, 2012 at 10:39:37AM -0400, Timothy Y. Chow wrote:
>> Buried in the now-defunct thread about fictionalism, Richard Epstein
>> wrote:
>>
>>> In my recent book *Reasoning in Science and Mathematics* (available from
>>> the Advanced Reasoning Forum) I present a view of mathematics as a
>>> science like physics or biology, proceeding by abstraction from
>>> experience, except that in mathematics all inferences within the system
>>> are meant to be valid rather than valid or strong. In that view of
>>> science, a law of science is not true or false but only true or false in
>>> application. Similarly, a claim such as 1 + 1 = 2 is not true or false,
>>> but only true or false in application. It fails, for example, in the
>>> case of one drop of water plus one drop of water = 2 drops of water, so
>>> that such an application falls outside the scope of the theory of
>>> arithmetic.
>>>
>>> On this view numbers are not real but are abstractions from counting and
>>> measuring, just as lines in Euclidean geometry are not real but only
>>> abstractions from our experience of drawing or sighting lines. The
>>> theory is applicable in a particular case if what we ignore in
>>> abstracting does not matter there.
>>
>> This sounds like a version of nominalism. On this view, I think,
>> mathematical nouns are akin to pronouns. So we can recognize the truth of
>>
>> You refer to me as "you" and refer to yourself as "me"
>>
>> while at the same time denying that asking whether "you" exists makes any
>> sense except insofar as it asks about the existence of some particular
>> *instantiation* of "you."
>>
>> This view must be very old, but as I think about it now, I don't recall it
>> being discussed explicitly very often. Can someone name some famous
>> proponents of it?
>>
>> Tim
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