Mathematical Realism and Mathematical Formalism
The differences in the philosophical views of mathematics can be calibrated to some extent by attitudes towards these new infinity.
Hilbert's views set him in complete opposition to Brouwer, another famous thinker, leading to a philosophical confrontation.
Hilbert's view that mathematics is a game that operates symbols purely according to certain rules is known as formalism.
This view does not necessarily prohibit the interpretation of this "formula game" as having one connection to reality or another, but in its basic form, it is less committed to the mathematical "entity" in question than the old forms of mathematical realism.
Plato's bust marble, 4th century
Hilbert's view that mathematics is a game that operates symbols purely according to certain rules is known as formalism.
This view does not necessarily prohibit the interpretation of this "formula game" as having one connection to reality or another, but in its basic form, it is less committed to the mathematical "entity" in question than the old forms of mathematical realism.
Such as Platonism (a view that naturally goes back to Plato, which holds that mathematical objects such as "1s" and "circles" do exist as persistent objects in a way that is independent of us and our understanding of them).
Brouwer's understanding of mathematics in a third way is very different from the two views above.
A modern statue of Gotlob Frege
One of Hilbert's most famous theorems, and at the heart of a deep disagreement between him and Brouwer, was what he called the cardinal point theorem.
The finer details don't matter: what interests philosophers, and what Blois opposes, is Hilbert's method of proving the theorem.
Hilbert's cardinal point theorem is an existential theorem – it is in the form of "at least one X".
When a mathematician proves that there is "at least one X", he can take one of two approaches: either prove how to find such an X, or prove that it is impossible to have no such X.
Hilbert's proof of the cardinal point theorem is unconstructive. Brouwer disagreed: he founded and passionately defended a method of mathematical philosophy known as intuitionism.
Bernardo Strozzi's Fables of Mathematics, 17th century
Intuitionism vs. Constructivism
Intuitionists reject mathematical objects as things that are not constructed by mental activity.
For Brouwer, there was a serious problem with the kind of non-constructed proofs technique that Hilbert used.
The school of mathematical philosophy that opposes these non-constructive proofs is known as constructivism.
Brouwer
Constructivists often oppose the existence of true infinity in mathematics, and this independent view is known as finitarianism (and its rather marginal cousin, ultra-finitarianism, which even opposes finite objects that are "too large to be reasonably constructed").
Thus, Hilbert and Brouwer, not only put forward different views on the reality and validity of mathematical objects, but also proposed very different mathematical methods.
Both gave birth to new studies of mathematical logic itself: the study of intuitionist logic, which did not exclude logical systems of intermediate laws, remains an active field of study today.
David Hilbert:
More famous, however, is Hilbert's early formalist approach, whose optimistic goal was to establish a system of axioms (axioms are initial statements that are always assumed to be true) from which all mathematics can be deduced and from which there is no contradiction in itself.
These concepts, referred to in mathematical logic as completeness and consistency, respectively, seem to be perfectly reasonable requirements for the chosen mathematical foundation.
Statue of Bolzano by Ernest Popper, 1849
In 1900, Hilbert published a list of 23 problems that he considered to be at the forefront of contemporary mathematics.
The second of these is to prove that his axioms of arithmetic are consistent.
This axiom system provides the basic arithmetic structures that we are familiar with – numbers, addition, subtraction, etc.
It is hoped that this set of axioms will be enough to formalize the rest of mathematics.
Plaque in honor of Gödel in Vienna
Gödel's Incompleteness Theorem: Trouble in Heaven
Kurt-Gödel's two incompleteness theorems are now famous, and they have proved that no axiom system that includes arithmetic can prove its own consistency, thus quelling the starry reading of Hilbert's plan.
These theorems are precise and subtle logical theorems, and philosophers have been cautious in considering their impact on mathematical realism (Gödel himself was still a staunch Platonist).
While Hilbert's plans weren't necessarily completely stalled after Gödel, these theorems were a watershed moment in mathematical logic, and they have been the subject of endless philosophical discussion ever since.
Hilbert's method is neither the first nor the last method on which mathematical axioms are based.
There were many grand plans at that time.
Frege, and later Russell, led the logicist approach, which aimed to reduce mathematical theorems to logical propositions.
Russell found a well-known serious problem in Frege's method, and one of his axioms was to allow the creation of a set by invoking a set of all things that satisfy a given property.
But this axiom falls into a contradiction, now known as Russell's paradox: this law allows all collections that do not contain themselves, i.e., a meaningless entity.
Gödel's theorem, in turn, seems to have put the brakes on Russell's own logicist ambitions, and mathematicians have switched to less ambitious approaches.
Both Frege and Russell themselves were integral to the early development of Ludwig Wittgenstein, whose work had a wide range of further influences on the philosophy of mathematics, including the place of logic and its relationship to natural language.
Bertrand-Russell's 1957 photo
Old Problems, New Problems: The Future of Mathematical Philosophy
Eventually, a workable solution to the axiomatic problem of set theory was found, which is the Zelmereau-Flekel axiom (and the axiom of choice, which has been controversial in history but is not very controversial today).
In practice, this ontology contains only one object, the set, from which everything is constructed, and is the "default" choice (though by no means the only one) for today's mathematicians.
Zemelo-Flekel set theory, which has gone all the way from philosophical speculation to concrete mathematical knowledge, is now itself a mathematical object of study by logicians.
However, just as Cantor's concept of sets challenged the way philosophers thought about mathematics, newer abstractions began to challenge the way philosophers thought about mathematics as new foundational methods emerged.
As the interaction between philosophy and mathematics deepens, not only are old problems still fresh, but new problems are emerging from new ideas in mathematics, keeping philosophers busy.
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