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 Posted: Sun Aug 28, 2005 2:41 am Post subject: Lakatos and Karl Popper |
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"John Jacobson aka Captain Jake" <jakeNOSPAM (AT) jsnewsreader (DOT) com> writes:
| Quote: | Lauchlan M
I don't see how you get to 'mathematics is just a language' from the thought
of Karl Popper.
Karl Popper: There is no such thing as proof, only disproof.
Therefore any "proof" offered by mathematics is tautological, simply a
matter of definition, nothing more. Matters of definition are language
issues.
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Not everyone accepts Popper's formulation
From the Stanford Encyclopedia of Philosophy
"Lakatos flatly denies that there are critical tests, in the Popperian sense,
in science, and argues the point convincingly by turning the above example of an
alleged critical test on its head. What, he asks, would have happened if Galle
had not found the planet Neptune? Would Newtonian physics have been abandoned,
or would Newton's theory have been falsified?
The answer is clearly not, for Galle's failure could have been attributed
to any number of causes other than the falsity of Newtonian physics (e.g.
the interference of the earth's atmosphere with the telescope, the existence
of an asteroid belt which hides the new planet from the earth, etc).
The point here is that the u2018falsification/corroborationu2019
disjunction offered by Popper is far too logically neat: non-corroboration
is not necessarily falsification, and falsification of a high-level scientific
theory is never brought about by an isolated observation or set of
observations. Such theories are, it is now generally accepted,
highly resistant to falsification.
They are falsified, if at all, Lakatos argues, not by Popperian critical tests,
but rather within the elaborate context of the research programmes associated
with them gradually grinding to a halt, with the result that an ever-widening
gap opens up between the facts to be explained, and the research programmes
themselves. (Lakatos, I. The Methodology of Scientific Research Programmes, passim).
Popper's distinction between the logic of falsifiability and its applied
methodology does not in the end do full justice to the fact that all
high-level theories grow and live despite the existence of anomalies
(i.e. events/phenomena which are incompatible with the theories).
The existence of such anomalies is not usually taken by the working
scientist as an indication that the theory in question is false;
on the contrary, he will usually, and necessarily, assume that
the auxiliary hypotheses which are associated with the theory
can be modified to incorporate, and explain, existing anomalies."
--
Seek simplicity and mistrust it.
Alfred Whitehead
A witty saying proves nothing.
Voltaire
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Lauchlan M
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| Posted: Sun Aug 28, 2005 4:46 am Post subject: Re: Lakatos and Karl Popper |
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| Quote: | I don't see how you get to 'mathematics is just a language' from the
thought
of Karl Popper.
Karl Popper: There is no such thing as proof, only disproof.
Therefore any "proof" offered by mathematics is tautological, simply a
matter of definition, nothing more. Matters of definition are language
issues.
|
Jake, this is simply wrong.
There are a whole range of complex issues touched on by this simplistic
statement of yours, and I haven't got the time and this is not the place to
go into it. I've set the follow-up to off-topic if you want to take it up.
But briefly, Popper noted an assymetry between proof and disproof in an
empirical context, while your statement about mathematical proof is in an
entirely different context, ie abstract and logical proof.
You are conflating two entirely different contexts, and what you say is
simply nonsense. A mathematical proof is a proof, regardless of what you
say. It is reasoning from premises to necessary conclusions given the
relevant rules of argument. _Given_ the definitions and premises, what
follows?
But that is more than language. It is supposition of certain kinds of
objects (eg numbers, or newtons laws of motion, or whatever) followed by
logical reasoning. It _uses_ a language but _is more than_ a language. Pure
mathematics, for example, refers to no real world objects but does refer to
abstract mathematical objects. It does have content.
HTH
Lauchlan M
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