Foundations

The foundations of mathematics – what are they?

We have to deal with a number of issues:

What is mathematics – and what is its place in the world… and the world of knowledge and language?

What is a foundation? Roughly, a foundation makes for security

What is the foundation of a fact. “Fact” has a number of meanings; here, a fact is a truth. The foundation of a fact is the basis of its truth. There are a number of ways a fact could have foundation. It could be so simple and plain to view that it could not be doubted. How might it be plain to view? Through the senses or through intuition. We would then have to ask, why the senses or the intuition provide a foundation. How can they provide a foundation and how does that provision obtain in the case of the particular fact. Alternatively, the fact could be the logical consequence of another fact. This does not eliminate the issue of the direct foundation of facts because of the question of the truth of the other fact. Additionally, the question of the foundation of “logical consequence,” i.e. of logic or inference arises

Two approaches to these questions arise. The first is that of a direct and positive foundation of “fact and context” as a whole. This requires a metaphysics. The second is “trial and error.” In fact, the reality is a combination. There is always some degree of system and it is not individual facts alone that are subject to scrutiny but, also, systems as a whole. In mathematics the systems are the mathematical disciplines: arithmetic, geometry, logic… What is a system? In the developed disciplines of mathematics two approaches to systematization are the axiomatic and the constructive

“The foundations of mathematics is the study of the logical and philosophical basis of mathematics, including whether the axioms of a given system ensure its completeness and its consistency. Because mathematics has served as a model for rational inquiry in the West and is used extensively in the sciences, foundational studies have far-reaching consequences for the reliability and extensibility of rational thought itself.”

What do we learn from the history of these topics?

Can we “draw it all together” – and, if so, how?

Logicism

In the latter part of the 19^th century Gottlob Frege developed, in his 1879 work Begriffsschrift, a theory of quantifiers in logic that was more or less adequate to the needs of a mathematical logic. Frege also expressed the view that the inferences of mathematics are based upon general laws of logic. This was expounded by Bertrand Russell in Principles of Mathematics published in 1903 and developed by Russell and Alfred North Whitehead in Principia Mathematica, 1910

Logicism, also called the Frege-Russell thesis, is the view that the concepts and truths of mathematics can be reduced to those of mathematics. Logicism is not generally accepted as a complete foundation of mathematics. For example, Kurt Gödel showed that no system such as the Principia Mathematica can be complete in the sense that it will answer all the questions of mathematics

So far nothing has been said about mathematics itself. There is sufficient doubt about the nature of that activity that there is the well known quip “mathematics is what mathematicians do” – motivated partly, no doubt, by a pragmatic desire to get on with mathematics rather than talk about it. Russell provided a definition: Pure mathematics is the class of all propositions of the form “p implies q,” where p and q are propositions, and neither p nor q contains any constants except logical constants. In view of the variety of mathematics, of the apparent lack of mathematical content of this formulation, and in view of the question why this formulation should exclude non-mathematical results – it can, of course, be regarded as a definition of mathematics but that is not necessarily definitive – the achievement of Principia Mathematica is remarkable. And, yet, the incompleteness results of Gödel stand in the way of logicism being a complete foundation of mathematics. Additionally, the development of logicism requires the theory of sets which necessitates axioms peculiar to itself – such as the axiom of infinity and the axiom of choice and the continuum hypothesis which latter two are independent axioms in the theory of sets – and are, only questionably, part of logic

Platonism

Note how the elements of ‘our’ understanding and being [and, therefore, all being] are falling together as one

If individuals are truly creative, then no simple form of Platonism can obtain. What is needed, then, is a Platonism that will include creative presence in the universe. Combine with comments below in Is mathematics independent of its empirical origins?

Think of geometry. It has to do with shapes in space and their properties and relationships. Think of the axiomatic system of Euclid. A fairly extensive body of knowledge and potential knowledge is reduced to a system of undefined terms, definitions, postulates and axioms – in modern thought the distinction between axioms and postulates vanishes. In geometry, it is as though there are universal forms that are being expressed. This is the Platonic view of mathematics and the foundation would be to find these forms

Formalism

Platonism is related to the formalism of David Hilbert which specified that in addition to logic – needed to establish theorems and consistency – mathematics requires its own special axioms and terms that are provided as abstract terms. The program of formalism was to demonstrate the consistency and completeness of the various axiom systems. A question is on what shall the completeness be based – how shall we know it. There are some subtle arguments here including modeling. Hilbert’s program of founding mathematics or parts of mathematics as formal systems is also subject to the Gödel incompleteness results. However, formalism, the idea that the axiomatic method may provide a foundation to mathematics continues to be, perhaps, the most useful practical yet rational foundation of mathematics

Intuitionism

L.E.J. Brouwer’s view was that mathematical truths are known in intuition. How can that be, given the esoteric nature of much of mathematics? First, the natural numbers can according to intuitionism be known intuitively. Although, originally, the natural numbers were derived from the world of experience, the individual has the ability to perform a sequence of mental acts – a first act, then another, then another and so on endlessly. Thus, one obtains a fundamental series, of which the best known is the series of natural numbers. The intuitionist program is to found mathematics upon the intuition of the natural numbers. Thus, “2 + 2 = 4” follows from performing mental constructions for 2 + 2 and 4 and showing that they are the same. Additionally, mathematics is not dependent on classical logic. For Brouwer, logic is founded and based on his mathematics – supported by the doing of mathematics. The rules for deriving new theorems from old are intuitively arrived at. These can be presented in the form of a symbolic logic called mathematical logic; this is a subdomain of mathematics and its use outside mathematics would be “senseless.”

Not only is mathematics not based on classical logic, the intuitive construction of the natural numbers and, so, mathematics is independent of language. Mathematics is self-generating, relying in no way on other philosophies or logic. The only role of the usual symbolic devices, including ordinary language, are for communication

Intuitionism is an austere view of mathematics. The law of the excluded middle is excluded as is the actual infinite

Wittgenstein’s views lend some support to intuitionism in that logic is discovered as a practical device rather than given as an eternal, universal Platonic system

Clearly, intuitionism is quite opposed to formalism and logicism. Without the attendant philosophy, the Principia Mathematica can be conceived as a purely axiomatic or formal development of the edifice of mathematics

Is not mathematics empirical?

I am not a Platonic Realist – that means I do not hold that there is a separate universe of Platonic forms. We may define or conceive of a [conceptual] space of ideas that include forms but that is a mental construct. There is one world, one universe

Let us equate that conceptual Platonic universe with the possibility of form in the creation from nothing. Then as we have seen in A Map of My World

The variety of that universe is practically infinite, not reducible to an algorithmic specification, does not ever exist as an actual universe

Therefore, for finite minds though not the mind of God, mathematics never gets out of its empirical origins

Think of the origin of mathematics – regardless of the actual nature of mathematics – surely the discovery was empirically based. Mathematics began as the mathematics of human beings. It is given, of course, that the discovers had or developed the ability to “see” the forms. In the beginning, science and mathematics were the empirical and the theoretical face of the same thing – the discovery of nature. the development of mathematics and the nature of its use encouraged separation. But we saw that to man, mathematics remains experimental. Platonism is an ideal. In reality the forms, at the present time, remain experimental in their discovery and creation

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