The Foundations of Mathematics
To understand this text, it might be helpful to keep track of some names; these are philosophers and mathematicians who have worked on the problems of logic and mathematics which are discussed below. In some cases, the boundaries are blurry between philosophers, mathematicians, and logicians.
Charles Parsons – professor at Harvard; here cited as a historian of logical and mathematical developments.
Gottlob Frege – professor in Germany; first to create a formal symbolic system, and the first to articulate the logicist point of view; he developed both propositional logic and quantified logic.
Bertrand Russell – professor in England; refined propositional and quantified logics; refined logicist thought; generally a successor to Frege in logicism.
W.V.O. Quine – professor at Harvard; worked within a logicist framework, rejecting some of Russell’s work and refining logicism; also worked within a set-theoretic framework.
Immanuel Kant – professor in Germany; viewed arithmetic as the result of synthetic a priori reasoning, and based upon the preconditions of space and time, which are, in turn, structures created by human reason; laid the foundations for the development of intuitionism.
Henri Poincaré – professor in France; wrote that mathematics was in no way based upon logic, but rather upon Kantian notions of space and time.
L.E.J. Brouwer – professor in the Netherlands; carried forward Poincaré’s Kantian intuitionism, and opposed Hilbert’s formalism.
David Hilbert – professor in Germany; axiomized mathematics and geometry; argued for formalism.
Georg Cantor – professor in Germany; essentially founded modern set theory.
Richard Dedekind – professor in Germany; refined set theory and number theory
Ernst Zermelo – professor in Germany; axiomized set theory; refined the concept of well-ordered.
Abraham Fraenkel – professor in Germany and Israel; refined Zermelo’s axiomization of set theory.
Kurt Gödel – professor in Germany and Princeton; showed that the axiom of choice does not impair the ZF system.
If you are reading this text and are not familiar with formal symbolic systems, please also consult the appendix at the end of this paper, where a brief overview of a propositional calculus is given.
Of the many topics which are of mutual interest to both mathematicians and philosophers, one is the foundation of mathematics: as Charles Parsons wrote, “foundational research has always been concerned with the problem of justifying mathematical statements and principles, with understanding why certain evident propositions are evident, with providing justification of accepted principles which seem not quite evident, and with finding and casting off principles which are
The Foundations of Mathematics – page 1 unjustified.”
Philosophers and mathematicians who work in this field are concerned to show the reliability of mathematics, that it indeed has a solid foundation, and that its results are therefore trustworthy. While this is regarded in ordinary life as an obvious truth – who ever really doubts that 7 + 5 = 12 ? – it becomes reasonable to investigate when mathematics produces results that are either counter-intuitive or beyond the possibility of confirmation or verification by ordinary sensory experience, or by clear a priori rational thought. Such attention-demanding results are common in both contemporary physics, but also in ancient riddles. Thus philosophers like Bertrand Russell hoped to prove, at least, the consistency of ordinary arithmetic; but universally-acknowledged, or even nearly-universally-acknowledged, results have eluded the philosophical and mathematical communities.
At stake is logical certainty and the degree to which we can rely on the results of reason: if we can’t take 7 + 5 = 12 as certain, what then could we take as certain? Charles Parsons notes that “two of the main qualities for which mathematics has always attracted the attention of philosophers are the great degree of systematization and the rigorous development of mathematical theories. The problem of systematization seems to be the initial problem in the foundations of mathematics, both because it has been a powerful force in the history of mathematics itself and because it sets the form of further investigations by picking out the fundamental concepts and principles. Also, the systematic integration of mathematics is an important basis of another philosophically prominent feature, its high degree of clarity and certainty. In mathematics systematization has taken a characteristic and highly developed form – the axiomatic method – which has from time to time been taken as a model for systematization in general.”
We can take a look at various groupings of philosophers and mathematicians: notably, the logicists, the intuitionists, the formalists, and the set-theoreticians. These are not the only four schools of thought regarding the foundations of mathematics, but they are four good examples with which to begin.
Logicism. - All mathematics, according to the logicist, are derived from logic. This belief is espoused by philosophers and mathematicians like Bertrand Russell and Gottlob Frege. A major text associated with the movement is Russell’s Principia Mathematica. Frege’s system was one of the first major attempts at a logicist foundation for mathematics. In addition to the usual axioms of logic, three other
The Foundations of Mathematics – page 2 axioms are added, in logicist foundations of mathematics, to make most of mathematical arithmetic possible; numerous other lesser axioms are added for creating various other parts of mathematics. The as the number of axioms increases, the elegance and plausibility of logicist program decreases. The three main axioms added are: the axiom of reducibility, the axiom of infinity, and the axiom of choice. The axiom of reducibility allows for a function of one type with variables of a different types. These three axioms, added to the usual axioms of propositional and quantified logic, and a few lesser axioms, then make the rest of mathematics derivable. - Thus math is founded on logic and proceeds forth from out of logic.
But the question arises, where is then the border between mathematics and logic? If math is founded upon logic, then all axioms belong to logic, but the three central axioms need for mathematics seem to be mathematical in nature: disguised mathematical postulates. So was math really drawn out of logic, or was math carefully hidden in logic by means of these axioms in order that it might seem that math would come forth form logic? So we ask, what is a mathematical axiom, what is a logical axiom? What is the border between math and logic? We must examine pure logic prior to the development of mathematical arithmetic, so that we can know truly what logic is, and how mathematics emerges from it. We must be sure that we have not used disguised mathematical axioms as logical axioms. Are the three axioms really logic, or merely disguised mathematics? What you count as logic determines whether or not logic can create mathematics. Frege, Russell, and others derived arithmetic from logic, but the question remains whether the non- arithmetical parts of mathematics can be additionally derived from logic. Can geometry and topology and probability be derived from the same logical basis which allegedly is the basis for arithmetic? Quine, the next generation of logicist, says yes, but many disagree with him. - Another question for the logicist, or another objection to logicism, asks why mathematics relates to the real world and the natural sciences, i.e., if logic generates mathematics, and logic is not a physical reality but rather a set of laws, prescriptive or descriptive, of thought, then why should mathematics bear any relation to physical reality? Why should math, if it is derived from thought, i.e., from logic, apply to anything but thought? Another way to ask this question is to ask whether math is self-referential or does it touch nature? The logicist must build a bridge from logic to nature to show how mathematics can be applied. Logicism also floundered on certain paradoxes, and the theory of types was introduced to remove those difficulties.
The Foundations of Mathematics – page 3 Intuitionism. - This view arose as an opposition to logicism. The intuitionists rely on direct intuition rather than on logic. This view is therefore somewhat Kantian, because Kant wrote that time and space are intuitions and he deduced arithmetic and geometry from intuitions – but the intuitionist school of mathematics goes beyond any questions posed by Kant – Poincaré was the main founder of this school – he said that axioms are not needed, rather we begin with our clear intuitions, e.g., we have clear intuitions of 1 and 0 and therefore we can build arithmetic. So we don’t need logical definitions. After Poincaré, Brower developed intuitionism and wrote that mathematics is a mental activity and the mind knows its own intuitions better than anything else. Numbers and arithmetic are built up from intuitions of plurality and time. Mathematics is thus synthetic, composing the truth out of parts. Logic belongs to language, not to mathematics, the and introduction of logic into mathematics creates errors. Mathematics belongs to the mind and as such is inexpressible or at best only partially expressible. Logic belongs to expression and hence is posterior to mathematics. Logic is founded upon mathematics; logic is the rules of language, language used to express mathematics – or language attempting to express the inexpressible mathematics. Neither proofs nor paradoxes are important, and there is no obligation to be logical. And consistency is not important. A less ambitious, minimalist logic, an intuitionist logic, is allowable, but systems like Russell’s are discarded. For example, the intuitionists allow the principle of the excluded middle to be used only in certain circumstances, in a certain narrow scope, i.e., finite cases. But intuitionists do not allow the use of the principle of the excluded middle in dealing with infinite sets, indeed intuitionists avoid the use of infinite sets as unintuitive. They also avoid certain types of numbers as unintuitive, e.g., transcendental numbers and π . Because they do not allow many logical and mathematical techniques, they do not accept many theorems and proofs of standard mathematics. Intuitionist mathematics are a minimalist mathematics lacking many features. They also reject sets defined by one property, and they demand that sets must be constructible. If I define a set as ∀(x) F(x) or as x∣F(x) the intuitionist claims that I do not have a clear intuition of what I am placing into my set and that I may be (invalidly in intuitionist eyes) working with an infinite set, or I may not even have have a set at all. (For some intuitionists, an empty set is not a set.) All mathematical entities must be either a direct intuition or a construction by intuitive means out of several intuitions. Hence intuitionists are either constructivists or are similar to constructivists. The drawback is that they have cut off so much of mathematics that what is left is of little use. By constructivist means they hope to reconstruct the theorems an proofs which they have destroyed (or disallowed) – but these
The Foundations of Mathematics – page 4 theorems and proofs are very abstract, removed from intuition, and even counterintuitive. So if they would rebuild the mathematics which they have destroyed, they would violate their own principles in doing so. What are then standards for truth? The intuition, clear and directly perceived by the mind, is the primary source truth for this group of mathematicians. Only certain logical, deductive, inductive, or algebraic operations or transformations are accepted by the intuitionists as being truly “salva veritae”. Therefore, the intuitionists do not accept negative proofs, i.e., a proof that N must exist because a contradiction arises from assuming that N does not exist.
Formalism. - This school arose against the intuitionists and was founded by Hilbert. The formalists write that logic operates on mathematics, that logic is the formal procedure, and mathematics is the content. Against the logicists, the formalists write that mathematics cannot be deduced from logic. If we view both logic and mathematics as symbols and symbol manipulation, then we are freed from many paradoxes, and we view them merely as uninterpreted symbols. For example, √-1 has no intuitive meaning, but we can use the uninterpreted symbol and create rules for its manipulation. Formalism has been characterized as certainty bought at the price of meaninglessness. To the formalist, then, mathematics and logic consist only of “well formed formulas,” axioms, deductions, theorems, etc., and there is little consideration of truth. Yet even for the formalist, it is necessary to show that arithmetic is consistent because higher mathematics is based upon it. Hilbert wants to make a basis or foundation on the consistency of arithmetic, and he wants to use a “finitary” logic to show this consistency. A finitary logic is basic and simple and designed to please the intuitionists. Once the consistency of arithmetic is proven, we can discard the finitary restrictions and use more traditional logics. Against Hilbert it may be objected that his mathematics had “no intelligible connection with the real world,” and therefore could not be justifiably used in the natural sciences. Russell objected that Hilbert’s metaphysics of existence was not determinant, because Hilbert wanted to write that what was not contradictory existed, but Russell wrote that there were numerous possible non- contradictory axiomatic systems which contradict each other. Hilbert can reply that Russell’s objections can be directed also against Russell’s own logicism. Russell had written that in logic or mathematics we never know about what we are talking. Hilbert’s differences with Brower and the intuitionists are in basic philosophical principles, and so rather than being described in arguments and objections and defenses, can better be described as a parting of the ways and the choosing of alternate assumptions, because Brower simply writes that mathematics means
The Foundations of Mathematics – page 5 something. Against Brower, Hilbert writes that if one accepts Brower’s intuitionism, one loses most of higher mathematics. Against Hilbert it is objected that formalism makes mathematics meaningless, but Hilbert defends against this object by writing that he insists on the meaninglessness of mathematics only while exploring its foundations, but that once we have settled the foundational issues, we may again impute meaning to higher mathematics when we apply them to the natural sciences: in application mathematics regains meaning. Brower says that mere consistency does not prove existence, and Kant agrees. Against Brower it may also be asked, if we are not satisfied with his mathematics, why should we be satisfied with his metamathematical justification of mathematics by means of finitary logic? If metamathematics is designed to justify mathematics, we can properly ask, why is Hilbert’s finitary metamathematics any more convincing than his mathematics? Yet Hilbert can muster some support for his foundations, because his is the most rigorous set of building blocks of which mathematics can be built. Compare Hilbert’s formalism to logicism: logicism admits that its axioms are arbitrary, so it has no claim over against formalism. And intuitionism has only its intuitions to offer as guarantee of the consistency. So all three schools – logicism, formalism, and intuitionism – each seek to find consistency, but none can provide a rigorous proof of consistency – so each group can offer only claims, but no more certainty than any other group.
Set Theoretic. - This school of mathematicians grew out of the work of Dedekind and Cantor, who began the axiomizing of sets. The first and majorly important system is the Zermelo-Fraenkel axioms, the consistency of which has not been proved. Against the set theorists it is written that they use the axiom of choice. Also the set theorists are not picky about which logic they use on sets. The axioms of set theory are artificial and non-intuitionistic. Against the set theorists it is written that if they use such non-intuitive axioms, then why do they not merely assume arithmetic? Quine is a set theorist.
The four main groups – logicists, intuitionists, formalists, and set theorists – all want the consistency of arithmetic at least, but none of them really ever proved it. Now, a “theorem proved” must be proved by four different standards of proof.
The Axiom of Choice. - Given any collection of sets, finite or infinite, one can select an object from each set and form a new set. The axiom of choice is needed to show that any set is well-ordered. Against the axiom of choice, it is written that no law specified which element was chosen from each set, and thus the axiom of
The Foundations of Mathematics – page 6 choice is to that extent arbitrary. In defense of the axiom of choice, we write that the choice is determined or stipulated. Again the axiom of choice, Russell writes that sets are defined by a property of the members – yet the axiom of choice is an arbitrary choice, not by properties. Russell is saying that the set created by the axiom of choice is not really a set at all. Zermelo defended the axiom of choice “in an intuitive sense.” He showed that the axiom of choice is the process of making something well ordered – that the axiom of choice is well ordering. He defended the axiom of choice by writing that it had not yet led to contradictions. Problem: is the possibility of choice enough? Or must the choice “exist”? Gödel showed that if ZF (the Zermelo-Fraenkel system) is consistent, then ZF with the axiom of choice is consistent. The axiom of choice cannot be proven from ZF. A set theorist then has options about which axiom base to use: (1) ZF with the axiom of choice, (2) ZF, (3) ZF with the continuum hypothesis, or (4) ZF with both the axiom of choice and the continuum hypothesis. Both the adoption and the denial of the axiom of choice have non-intuitive technical results.
Starting with Gottlob Frege, the development of formal symbolic systems, and their axiomization, has been important for the development of logic and mathematics. Such systems reduce ambiguity, and make notions such as validity, correctness, completeness, and truth calculable. One of the simplest and most common such systems is the propositional calculus, developed by Bertrand Russell, and refined by others since.
We begin by defining an alphabet. We will use letters p q r s as primitive symbols to represent propositions. We will use ~ and ⋁ as primitive symbols to represent operators.
Secondly, we establish formation rules to govern the creation of well-formed formulas. A well-formed formula is often called a “wff” and is produced when our primitive symbols are combined according to the formation rules. We stipulate, for example, that p ⋁ q and ~p are well-formed, but pq and ~⋁ are not.
We may later introduce addition operators and define them in terms of the original operators stipulated in the first creation of the system. Let us add the operator ⋀ and define it such that p ⋀ q is equivalent to ~(~p ⋁ ~q).
We may choose to interpret our symbols intuitively, or we may leave them as meaningless. The logic will proceed on its own, unaffected by any interpretations or meanings we may give to the symbols. It is convenient, however, to refer to ~ as negation, and to ⋁ as a an exclusive disjunction (not to be confused with an inclusive disjunction). Thus, from our definition above, p ⋀ q is read as “p and q” and ~(~p ⋁ ~q) is read as “it is not the case that either not p or not q”.
A well-formed formula is said to be “valid” if it is true by nature of its form; it is a tautology. Examples include:
p ⋁ ~p
By contrast, a well-formed formula is “unsatisfiable” if it is false by nature of its form:
~p ⋀ p
Most formulas are neither tautologies nor unsatisfiable. They are valid if they are well-formed and if, given the
The Foundations of Mathematics – page 7 truth-values of the individual propositional variables in them (p q r s), the formula as a whole is true. A procedure known as a “truth table” is often used to judge validity.
We will see that equivalent forms may be substituted for each other, by definition:
p ⋀ q
~(~p ⋁ ~q)
Given the possibility of such substitutions, and of the creation of additional operators which can then be defined in terms of the two primitive operators, it is clear that there can be many equivalent expressions. When it is necessary to distinguish between equivalent expressions, which are logically indistinguishable, but typographically divergent, we may stipulate a “conjunct normal form.”
We may select a group of well-formed formulas to be axioms by stipulation. We then form transformation rules which instruct us about how to use the axioms. Generally, transformation rules begin with “uniform substitution” and “modus ponens” as primitive; more transformation rules can be generated out of them. We may not, however, add to, or change, our axioms.
We now have a logical system, a propositional calculus. If we desire to attach interpretations to the propositional variables, we will choose axioms which are valid, and transformation rules which preserve validity. The well- formed formulas which we obtain by applying the transformation rules are known as “theorems,” and a “thesis” is a well-formed formula which is either an axiom or a theorem of the system.
An axiomatic system is said to be “consistent” if, and only if, no thesis is the negation of any other thesis, i.e., a proposition and its negation cannot both be formed using the axioms and transformation rules (logicians often use “iff” to abbreviate “if and only if”).
There are two further senses of “consistent” - first, a system can be consistent if and only if no well-formed formula consisting of a single propositional variable is a thesis; second, a system can be consistent if and only if not every well-formed formula is a thesis.
A system may be weakly complete, which is to say that every valid well-formed formula is of the system is derivable as a thesis. Under interpretation, this is to say, all truths can be proven.
A system may be strongly complete, which is to say that it cannot have any more theses that it has without falling into inconsistency. Because there are three types of consistency, there are three types of strong completeness.
The Foundations of Mathematics – page 8