Mathematical knowledge is ultimately obtained from what follows from mathematical principles.
Mathematical model, either a physical representation of mathematical concepts or a mathematical representation of reality.
Some readers may find the mathematical chapters heavy going, but Mr Derbyshire makes a valiant attempt at explaining the mathematical ideas around the problem.
It covers a broad spectrum of subjects: algebra and logic, real and complex analysis, functional analysis, differential equations, mathematical physics, geometry and topology, probability and mathematical statistics, mathematical cybernetics, mathematical economics, mathematical problems of geophysics and tomography, numerical methods, and optimization theory.
A related benefit of platonism is that it allows one to take mathematical discourse literally, given that mathematical terms refer.
So, the question about the provability or unprovability of any given mathematical statement becomes a sensible mathematical question.
But we can turn the tables around: mathematical necessity is unrestricted and false mathematical theories are just impossible theories.
In one respect, mathematical fictionalists offer a uniform semantics for mathematical and scientific discourse, in another respect, they don't.
So the problem of how to balance mathematical and scientific standards is particularly pressing for the mathematical-cum-scientific naturalist.
As just noted, without such operators, mathematical fictionalism produces non-standard attributions of truth-values to mathematical statements.
Given that mathematical objects do not exist, on the mathematical fictionalist perspective, the problem of how we can obtain knowledge of them simply vanishes.
In an attempt to agree with the truth-value assignments that are usually displayed in mathematical discourse, the mathematical fictionalist introduces a fictional operator: ‘According to mathematical theory M…’.
It is here that revolutionary fictionalism comes to the rescue: so long as there is some worthwhile aim of mathematical discourse despite the nonexistence of mathematical entities, mathematical discourse need not be abandoned.
The epistemological problem of mathematics is the problem of explaining the possibility of mathematical knowledge, given that mathematical objects themselves do not seem to play any role in generating our mathematical beliefs (Field 1989).
The fact that we regard mathematical propositions as being about mathematical objects and mathematical investigation “as the exploration of these objects” is “already mathematical alchemy”, claims Wittgenstein (RFM V, §16), since
On the face of it, Hoare seems to be committed to what we shall call the mathematical perspective, i.e., that correctness is a mathematical affair; i.e., establishing that a program is correct relative to a specification involves only a mathematical proof.
Given Wittgenstein’s rejection of infinite mathematical extensions, he adopts finitistic, constructive views on mathematical quantification, mathematical decidability, the nature of real numbers, and Cantor’s diagonal proof of the existence of infinite sets of greater cardinalities.
The difference between the ‘anthropological’ and the mathematical account is that in the first we are not tempted to speak of ‘mathematical facts’, but rather that in this account the facts are never mathematical ones, never make mathematical propositions true or false.
If every consistent mathematical theory is true of some universe of mathematical objects, then mathematical knowledge will, in some sense, be easy to obtain: provided that our mathematical theories are consistent, they are guaranteed to be true of some universe of mathematical objects.
A structure is a set of elements on which certain operations and relations are defined, a mathematical structure is just a structure in which the elements are mathematical objects (numbers, sets, vectors) and the operations mathematical ones, and a model is a mathematical structure used to represent some physically significant structure in the world.
- beyond question
Example: a mathematical certainty
- of or pertaining to or of the nature of mathematics
Example: a mathematical textbook
- statistically possible though highly improbable
Example: have a mathematical chance of making the playoffs
- relating to or having ability to think in or work with numbers
- characterized by the exactness or precision of mathematics
Example: mathematical precision
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A structure is a set of elements on which certain operations and relations are defined a mathematical structure is just a structure in which the elements are mathematical objects numbers sets vectors and the operations mathematical ones and a model is a mathematical structure used to represent some physically significant structure in the world