Philosophy of mathematics

This means that the definitions must be absolutely unambiguous and the proofs must be reducible to a succession of applications of syllogisms or inference rules,[a] without any use of empirical evidence and intuition.

[12] For example, the perihelion precession of Mercury could only be explained after the emergence of Einstein's general relativity, which replaced Newton's law of gravitation as a better mathematical model.

A notable example is the prime factorization of natural numbers that was discovered more than 2,000 years before its common use for secure internet communications through the RSA cryptosystem.

philosophers of mathematics aim to give accounts of this form of inquiry and its products as they stand, while others emphasize a role for themselves that goes beyond simple interpretation to critical analysis.

This perspective dominated the philosophy of mathematics through the time of Boole, Frege and Russell, but was brought into question by developments in the late 19th and early 20th centuries.

Three schools, formalism, intuitionism, and logicism, emerged at this time, partly in response to the increasingly widespread worry that mathematics as it stood, and analysis in particular, did not live up to the standards of certainty and rigor that had been taken for granted.

Surprising and counter-intuitive developments in formal logic and set theory early in the 20th century led to new questions concerning what was traditionally called the foundations of mathematics.

In his seminal Die Grundgesetze der Arithmetik (Basic Laws of Arithmetic) he built up arithmetic from a system of logic with a general principle of comprehension, which he called "Basic Law V" (for concepts F and G, the extension of F equals the extension of G if and only if for all objects a, Fa equals Ga), a principle that he took to be acceptable as part of logic.

Hilbert was initially a deductivist, but, as may be clear from above, he considered certain metamathematical methods to yield intrinsically meaningful results and was a realist with respect to the finitary arithmetic.

Other formalists, such as Rudolf Carnap, Alfred Tarski, and Haskell Curry, considered mathematics to be the investigation of formal axiom systems.

From this springboard, intuitionists seek to reconstruct what they consider to be the corrigible portion of mathematics in accordance with Kantian concepts of being, becoming, intuition, and knowledge.

Attempts have been made to use the concepts of Turing machine or computable function to fill this gap, leading to the claim that only questions regarding the behavior of finite algorithms are meaningful and should be investigated in mathematics.

Another variant of finitism is Euclidean arithmetic, a system developed by John Penn Mayberry in his book The Foundations of Mathematics in the Theory of Sets.

[45] Mayberry's system is Aristotelian in general inspiration and, despite his strong rejection of any role for operationalism or feasibility in the foundations of mathematics, comes to somewhat similar conclusions, such as, for instance, that super-exponentiation is not a legitimate finitary function.

Like nominalism, the post rem approach denies the existence of abstract mathematical objects with properties other than their place in a relational structure.

Embodied mind theories hold that mathematical thought is a natural outgrowth of the human cognitive apparatus which finds itself in our physical universe.

The cognitive processes of pattern-finding and distinguishing objects are also subject to neuroscience; if mathematics is considered to be relevant to a natural world (such as from realism or a degree of it, as opposed to pure solipsism).

Its actual relevance to reality, while accepted to be a trustworthy approximation (it is also suggested the evolution of perceptions, the body, and the senses may have been necessary for survival) is not necessarily accurate to a full realism (and is still subject to flaws such as illusion, assumptions (consequently; the foundations and axioms in which mathematics have been formed by humans), generalisations, deception, and hallucinations).

Alternatively, computer programmers may use hexadecimal for its 'human-friendly' representation of binary-coded values, rather than decimal (convenient for counting because humans have ten fingers).

The Euclidean arithmetic developed by John Penn Mayberry in his book The Foundations of Mathematics in the Theory of Sets[45] also falls into the Aristotelian realist tradition.

Mayberry, following Euclid, considers numbers to be simply "definite multitudes of units" realized in nature—such as "the members of the London Symphony Orchestra" or "the trees in Birnam wood".

Whether or not there are definite multitudes of units for which Euclid's Common Notion 5 (the whole is greater than the part) fails and which would consequently be reckoned as infinite is for Mayberry essentially a question about Nature and does not entail any transcendental suppositions.

John Stuart Mill seems to have been an advocate of a type of logical psychologism, as were many 19th-century German logicians such as Sigwart and Erdmann as well as a number of psychologists, past and present: for example, Gustave Le Bon.

He proposed that an alien species doing mathematics might well rely on quasi-empirical methods primarily, being willing often to forgo rigorous and axiomatic proofs, and still be doing mathematics—at perhaps a somewhat greater risk of failure of their calculations.

Quine suggests that mathematics seems completely certain because the role it plays in our web of belief is extraordinarily central, and that it would be extremely difficult for us to revise it, though not impossible.

Mathematical fictionalism was brought to fame in 1980 when Hartry Field published Science Without Numbers,[54] which rejected and in fact reversed Quine's indispensability argument.

Another fictionalist, Mary Leng, expresses the perspective succinctly by dismissing any seeming connection between mathematics and the physical world as "a happy coincidence".

Major discoveries can be made in one branch of mathematics and be relevant to another, yet the relationship goes undiscovered for lack of social contact between mathematicians.

Social constructivists argue each speciality forms its own epistemic community and often has great difficulty communicating, or motivating the investigation of unifying conjectures that might relate different areas of mathematics.

One line of defense is to maintain that this is false, so that mathematical reasoning uses some special intuition that involves contact with the Platonic realm.