[FOM] Are proofs in mathematics based on sufficient evidence?

[Charles Silver]

	  Tangentially, Richard L. Epstein has a critical thinking book,  
workbook, teacher's edition, CD (I think), in which, after  
establishing examples of deductively valid arguments in English, then  
presents a myriad of examples of arguments that are not deductively  
valid (like those we encounter daily), but are evaluated in terms of  
their strengths.  Some may be (from my weak memory) extremely strong:  
very, very strong; very strong; strong; not very strong; etc., etc.   
One can consider the arguments at the top of the strength list to be  
"inductively valid," or close to it--whatever that means.  Since these  
are everyday arguments, the non-deductively valid ones that are super- 
strong are very valuable outside of the realm of pure logic.  He  
provides answers to exercises, which he invites the reader to disagree  
with.  I regard some of the examples given on this thread to appear  
among his close-to-deductively-valid arguments.  (This is not an  
ordinary "critical thinking" book.  For one thing, Dick is a recursion  
theorist and has also written a book on computability.)

On Jul 16, 2010, at 10:06 PM, Vaughan Pratt wrote:

>
>
> On 7/8/2010 4:18 AM, Arnold Neumaier wrote:
>>
>> I wonder why you put mathematical proof and logical proof into the  
>> same
>> category, as opposed to legal or other kinds of proofs.
>>
>> There are worlds between these two notions of proof, in spite of the
>> common ground these notions have.
>
> (A belated response to Arnold's early response to my original  
> question.)
>
> The Wikipedia article Proof (truth) actually does distinguish these,  
> as
> it has done since I first wrote it.  Part of the confusion arose  
> when a
> zealous editor deleted all the material on both mathematical and  
> logical
> proof on the ground that only informal proof took evidence for  
> premises.
>  This editor did not see enough difference between the two to treat
> them differently in that regard.
>
> While there are plenty of nuanced distinctions, I see two somewhat
> independent binary distinctions in the concept "sufficient evidence  
> for
> truth."  (Not everyone will see the same ones, and I may change my own
> mind about these later on.)
>
> 1.  The evidence may be drawn either from experience or hypothesis.
>
> 2.  Sufficiency may be either soft or hard.
>
> 1.  Experiential evidence comes from nature, namely the real or
> sensorially apprehended world, augmented with inferences from that
> evidence about nature.  This includes the actual state of a computer
> gate, register, or memory.
>
> Hypothetical evidence comes either directly from what-if  
> counterfactuals
> or axioms or indirectly as consequences of direct hypotheticals
> (reasoning).  This includes the activity of program verification.
>
> One might call these respectively fact (real truth) and fiction
> (imagined truth).
>
> There is a phenomenon whereby fiction appears as fact: just as a
> pot-boiler that can't be put down conjures up images hard to  
> distinguish
> from facts gleaned from newspapers, so can mathematical axioms seem  
> real
> to the mathematician accustomed to intensively visualizing abstract
> universes.
>
> 2.  The soft-hard dichotomy in sufficiency is to me the same as the
> informal-formal dichotomy (I could be talked out of this but first  
> read
> the next paragraph).  Hard is when there are precise criteria that
> evidence must meet to constitute a complete proof, soft is  
> everything else.
>
> The term "precise" offers a loop-hole here.  Precision can only be
> measured up to some standard of equality, isomorphism, equivalence, or
> whatever.  Each such standard may have a mathematically or
> scientifically rigorous definition, but there may be more than one,  
> and
> they may induce a partial order on the standards.  We see this in  
> proof
> theory, with Girard's notion of proof net as an abstraction of
> sequential proof ("bureaucracy" to use Girard's term), and with the  
> even
> more abstract notion of proof contemplated in Dosen and Petric in  
> their
> 2004 book Proof-Theoretical Coherence, where a proof is simply a
> morphism interpreted as a proof in a category with suitable structure
> supporting that interpretation.
>
>
> These two distinctions combine in the following four ways, with the
> associated applications.
>
> Experience/soft    Scientific investigation, arguments in court, work,
> bars, home, etc.
>
> Hypothesis/soft    Mathematical reasoning, counterfactual reasoning
>
> Experience/hard    Formal deduction applied to the real world, whether
> it be Aristotle's syllogisms as popularized by Lewis Carroll, Boolean
> logic applied to database search, etc.
>
> Hypothesis/hard    Formal deduction applied to mathematics (the core
> focus of FOM perhaps?), but also to counterfactual reasoning about
> real-world situations.
>
>
> One distinction this analysis does not make is between counterfactuals
> about real-world situations and mathematical theories.  There may well
> be such a distinction in most people's minds; in that respect I may be
> out of step with everyone else.  To me every mathematical theory  
> *could*
> be about some real-world situation suitably abstracted.  One cannot
> reason about counterfactuals with every detail filled in, how would  
> you?
>  Unlike experiential evidence, counterfactual evidence can't be poked
> around in because there is no real world backing it up.  It is  
> therefore
> necessarily abstract.  One might quibble as to whether that  
> abstraction
> is like mathematical abstraction, but I have great difficulty in  
> drawing
> that line and therefore no basis for joining such a quibble.
>
> Vaughan Pratt
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