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la_flash
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Location: Konoha library
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 Posted: Thu Aug 17, 2006 1:26 am Post subject: Is mathematics absolute? |
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There has been debates regarding the absolutism of mathematics.
What do you think? Is mathematics really absolute?
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Ischaramoochie
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| Posted: Thu Aug 17, 2006 3:17 pm Post subject: |
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only in the context of its sphere of applicability, detemined by the axioms it employs. for instance, euclidian geometry would make little sense when used on the surfaces of spheres. ;)
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"The grandeur of a philosophy does not certify its truth."
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la_flash
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| Posted: Fri Aug 18, 2006 7:47 am Post subject: |
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^^ Quite true... and I agree with that...
The ff excerpts (and I forgot where did I get it) offers some insights re: absolutism of mathematics. I already posted this at PEX.
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The fundamental problem of the philosophy of mathematics is that of the status and foundation of mathematical knowledge. What is the basis of mathematical knowledge? What gives it its seeming certainty, and is this certainty justified? Two main currents in the philosophy of mathematics can be distinguished. These may be termed absolutist and conceptual change philosophies of mathematics, following the usage of Confrey (1981). Absolutist philosophies of mathematics, including Logicism, Formalism, Intuitionism and Platonism, assert that mathematics is a body of absolute and certain knowledge. In contrast, conceptual change philosophies assert that mathematics is corrigible, fallible and a changing social product. This second claim is shocking, for mathematics is seen by many to be the last bastion of certainty. Perhaps the most important statement of this claim is to be found in Lakatos (1976), and even here the editors added footnotes repudiating Lakatos' fallibilist philosophy of mathematics. Thus, it must be acknowledged that this is a controversial dichotomy. Whilst in science absolutist views have largely given way to conceptual change views, following the work of Hanson, Kuhn, Lakatos, Feyerabend and others, absolutist philosophies of mathematics are still the dominant view. Absolutists believe that mathematical truths are universal, independent of humankind (mathematics is discovered, not invented), and culture- and value-free.
However, the absolutist view is increasingly subject to challenge and attack, for example by Lakatos (1976, 1978), Davis and Hersh (1980), Kitcher (1983), and Tymoczko (1986), as well as many others including Putnam, Bloor and Wittgenstein (1956). The fallibilist position is gaining acceptance year by year, as is illustrated by the publications of philosophically minded mathematics educators in journals such as For the Learning of Mathematics. In the brief space available I will sketch two criticisms of absolutism.
Proof, via deductive logic, is the means by which the certainty of mathematical knowledge is established. However, absolute certainty cannot be gained in this way. As Lakatos (1978) shows, despite all the foundational work and development of mathematical logic, the quest for certainty in mathematics leads inevitably to an infinite regress. Any mathematical system depends on a set of assumptions, and there is no way of escaping them. All we can do is to minimise them, to get a reduced set of axioms and rules of proof. This reduced set cannot be dispensed with, only replaced by assumptions of at least the same strength. Thus we cannot establish the certainty of mathematics without assumptions, which therefore is conditional, not absolute certainty. Only from an assumed basis do the theorems of mathematics follow.
Given that mathematical knowledge is tentative in this sense, are not the theorems of mathematics certain within the assumed axiomatic system? Again the answer is negative. For to establish that mathematical systems are safe (ie. consistent, and we cannot have certainty without consistency), then Godel's Second Incompleteness Theorem shows that for any but the simplest systems (e.g. weaker than Peano Arithmetic) to prove consistency we must add to the set of assumptions, i.e. rely on the consistency of a larger set of assumptions. Thus any attempt to establish the certainty of mathematical knowledge via deductive logic and axiomatic systems fails, except in trivial cases, but including Intuitionism, Logicism and Formalism.
Disposing of absolutism is all very well, but a replacement philosophy must account for the unique features of mathematical knowledge. In particular: How to account for the apparent certainty and objectivity of mathematical knowledge? How, in Wigner's phrase, to account for 'the unreasonable effectiveness of mathematics' in describing the world, and indeed via science, in giving us an unparalleled power over the natural world?
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Ischaramoochie
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| Posted: Fri Aug 18, 2006 3:37 pm Post subject: |
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i'd have to agree with Einstein in his quote that "Insofar as the laws of mathematics describe reality, they are not certain; and insofar as they are certain, they do not describe reality."
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"The grandeur of a philosophy does not certify its truth."
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...Veritas? Quid est Veritas? |
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la_flash
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| Posted: Mon Aug 21, 2006 4:07 am Post subject: |
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^^ Hmm... I second the motion.
Do you agree that using deductive logic, one can not be certain as we can not escape infinite regress?
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rickym
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Joined: 12 Aug 2006
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| Posted: Mon Jun 23, 2008 5:04 pm Post subject: |
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| how about imaginary and complex numbers are they considered absolute? |
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