WITTGENSTEIN’S PHILOSOPHY
OF MATHEMATICS
Hazhir Roshangar
NOV 2019
introduction
• More than half of W.’s writing from 1929 to 1944 is
devoted to mathematics.
• His chief contribution: philosophy of mathematics.
• Platonism (mathematical objects are real)
• Logicism (mathematics is reducible to logic)
• Two beliefs that W. maintains:
• mathematical propositions are not real propositions,
• and that mathematical truth doesn’t refer to any object.
• W. belongs to no school.
• His job was to clear the misconceptions.
early period (tractatus)
• Not a logicist: a mathematical identity is not stating a truth
at all (it is a rule).
• tautologies are not about reality > they do not express
truth-value.
• Mathematical ‘pseudo-propositions’ are equations: things
with same meanings. (They say what they show).
• equation and substitution.
• We prove : ‘seeing’ that two expressions have the same
meaning.
• All by symbol manipulation.
• W. uses arithmetical identities to show that mathematics is
simply a machine for tautologies.
early period (tractatus)
• numerals are not names of objects, but labels
marking the number of the iterations of a
process of proposition derivation. > the
natural numbers represent stages in the
execution of a logical operation.
• The general form of an operation Ω′ η is
ξ, 𝑁 ξ ′ η = η, ξ, 𝑁 ξ .
early period (tractatus)
• W. introduces [𝑎, 𝑥, 𝑂′𝑥] as the general term of
series of forms, e.g., (𝑎, 𝑂′𝑎, 𝑂′𝑂′𝑎); and all the
general forms that he introduces in TLP (of an
operation, a proposition, and a natural
number,) are modeled on this form.
• The first term is the beginning of a series, the
second one is an arbitrary term from the list,
and 𝑂’𝑥 is the term that immediately follows 𝑥
in the series.
early period (tractatus)
• All natural numbers can be generated by repeated
iterations of the general form of a natural number:
[0,ξ,ξ+1].
• “[m]athematics is a method of logic”.
• general form of a number ([0,ξ,ξ+1]) and the
general form of a proposition ( 𝑝, 𝜉, 𝑁 𝜉 ) are both
the same thing: general form of a formal
operation.
• 𝑝 stands for all elemental propositions | 𝜉 is any set of
propositions | 𝑁 𝜉 stands for the negation of all members of 𝜉.
early period (tractatus)
• mathematical inferences are just instances of
application of logical rules (by the symbol
alone) without any observations of states of
affairs.
middle period
• Differences From the Early Period
• In the Tractatus, W. was influenced by Russell
and Frege, > Brouwer, Weyl, Hilbert, and
Skolem.
• The most important difference : rejects
quantification over infinite domains. So,
propositions can no longer be infinite
conjunctions and disjunctions : there is no
infinity.
middle period
• We Invent the Mathematical Machine
• we invent mathematics and we expand it by
calculation and proof.
• in mathematics we use numbers, we don’t talk
about them.
• For example, looking at the schema |||| : arithmetic
doesn’t talk about the lines, it merely operates with
them.
• mathematics is a machine that uses these kinds of
symbols for its unfolding.
middle period
• These symbols don’t have any inherent
meaning by themselves.
• “However queer it sounds, the further expansion of an irrational
number is a further expansion of mathematics”.
• Any meaning in this regard simply arises from
the process of doing these kinds of jobs.
• For example, to know the meaning of 2 + 2 = 4,
(process of calculation is what is essential).
middle period
• How a proposition is verified is what it says
(meaning).
• Mathematical symbols lack (objective)
meaning, they are not proxy for (concrete)
things which are their meanings.
• Therefore, in mathematics everything is
algorithm and nothing is meaning.
middle period
• The Infinite
• The infinite is not a quantity and mathematical
infinite resides in recursive rules.
• An irrational number is not something that we
can find > it’s the rule itself (same goes for a
geometric line). > a never-ending iteration.
• He adopts the radical position that all expressions
that quantify over an infinite domain are
meaningless: (Goldbach’s Conjecture, the Twin
Prime Conjecture, Euclid’s Prime Number
Theorem).
middle period
• Proof by induction: Inductive Base: 𝑃(1), Inductive Step:
𝑃(𝑛) ⇒ 𝑃(𝑛 + 1) therefore we can say 𝑃(𝑚).
• W. only accepts induction in a restricted sense: e.g., we have
700 modus ponens to prove via the Inductive Step, and
induction can save us time.
• It doesn’t permit us to go on to infinity. Inductive Step proxy
for a more direct proof.
• Extensions and Intentions
• W. thinks that mathematics consists of extensions and
intentions (i.e., rules), so an irrational is only an extension
as long as we consider the numeral itself e.g., ‘√2’.
late period
• Lack of Systemic Theses and Illuminating Effect of Philosophy and
the Infinite
• W.’s later philosophy is its lack of theses. (meta-level advice to find
the correct account, rather than developing the account.)
• philosophical light could get rid of excessive mathematical
branches.
• An important constant in Wittgenstein’s Philosophy of Mathematics
(middle and late,) is that he consistently maintains that mathematics
is our, human invention.
• “chess only had to be discovered, it was always there!”
• The later Wittgenstein still rejects the actual infinite and infinite
mathematical extensions. However there maybe more room for the
“potential infinite”.
late period
• Arbitrary Formal Games and Mathematics
• As we said, mathematical identities like ‘3 × 3 = 9’ are rules. Thus,
for us, the question is whether they are arbitrary or not?
• But, such rules stem from empirical experiences, and thus they
are not arbitrary. We have mathematics because at some point
empirical regularities have “hardened” into rules.
• These rules then become a paradigm, that experience will be
judged against it.
• So, when the arithmetical rule is formed, not only it no longer
requires empirical experience for its justification, but the empirical
experience itself will be justified by its means.
• mathematical propositions are non-revisable in light of empirical
investigation. In his own words: “mathematics as such is always
measure, not thing measured”.
late period
• The Platonism Access Problem
• To address the Platonism Access Problem W. abandons the assumption
that knowledge is made by a causal interaction with the known objects.
• W. thinks that knowledge itself is a family-resemblance concept.
• Further, he holds that this type of thinking in which knowledge is made by
causal interaction with reality is itself based on the epistemological
context of natural sciences.
• So, by ‘transferring’ the methodology of natural sciences assuming that
mathematics has the same kind of reality, we have already assumed too
much and misled our philosophy.
• Here, what W. rejects is not the possibility to talk about mathematical
objects and truths, rather it is that we cannot hope for the same kind of
justification techniques as that of empirical sciences.
• Therefore, we can be free of an ‘illegitimate question’ which is trying to
pinpoint the objects of mathematical discourse.
late period
• Constructivism and the Infinite
• When W. rejects the notion of actual infinity (and state the
meaninglessness of adherence to infinity for example in the
case of disputing the validity of irrational numbers or that
of inductive proofs) or the notion of existence proofs, we
may label him as a finitist-constructivist.
• But if we get rid of infinity we can end up disputing large
portions of current mathematics, for example, real analysis.
• However, W. always held that philosophy may not interfere
with the actual usage of a language and that by philosophical
endeavor we may only hope for describing a language.
late period
• Our Mathematical Practice and Our Nature
• Understanding the importance of the empirical regularities for
mathematics amounts for Wittgenstein to a “silent revolution”, in
addition to the well-known “overt revolution” (the repudiation of the
Tractatus).
• Some intuitionists (who reject the logical rule of excluded middle) such
as Brouwer see mathematics as a languageless activity of mind.
• W. sees mathematics inseparable from the human language and
practice. The meaning of calculations and proofs is in our practices.
• W. says that we know as much mathematics as God does.
• There is no transcendent justification for what we do in mathematics. For
example, mathematical identities are codifications of contingent but
very robust method of correspondence between the collective human
mind and the regularities of the objective world.
late period
• Humans have a sort of a behavioral agreement
when doing mathematics.
• This type of agreement consists in all of us having,
roughly, the same natural reactions when
presented with the same ‘mathematically’ related
situations (arranging, sorting, recognizing
shapes, and so forth). So:
• we can be trained to have certain similar reactions (neuro-physiological
basis) e.g., in doing a multiplication,
• the world itself presents a certain stability, many regular features.
late period
• Mathematics is objective (based on regularities),
but its objects are not transcendent. This is very
clever because it avoids the problems of both
Platonism and subjectivism.
• For W. A list of strokes is not an abstract list, it is
concrete—it is concretely expressed on a medium
(ink on paper). For example, ||| can be thought as a
paradigm for the number 3. Further, the meaning
of these strokes, arises in our practices.
late period
• Numerals outside mathematics are being used transitively, they derive
their meaning from paradigmatic samples. Numerals within
mathematics express internal relations between different samples.
• In PI, W. notes that any process can be made out to accord with some
rule, and thus no process could be determined by a rule (the rulefollowing
paradox).
• This is because it’s impossible to make a list that contains all the rules. But
W. argues that there is something related to our nature, and how the
mind works that make us able to follow rules as it is intended.
• The behavioral agreement is not simply an agreement of opinion. > the
whole natural procedure that leads to an agreement about a matter at
hand. It is a ‘consensus of action’. E.g., if we are to run a simulation…
• In W.’s own words these agreements are “not agreement[s] in opinions
but in form of life.”
late period
• Proof As the Meaning of a Mathematical Proposition
• Meaning of mathematical proposition is its proof.
• E.g., Goldbach’s Conjecture that is justified heuristically (brute force or
statistical methods) implies a different meaning from when it is rigorously
proven.
• A mathematical proof of a proposition shows the internal relations of
the proposition to a system of mathematical rules.
• A proof is like a cinematographic picture.
• W. holds that a completely analyzed mathematical proposition (i.e. when
all the internal relations are depicted) is its own proof.
• If the meaning of an arithmetical generalization is given by its proof, then
none of us understands Goldbach's conjecture. But then how could
anyone try to find one? But W. mentions that a new problem (a senseless
conjecture) can stimulate mathematicians to try to find some internal
relations, and there is nothing wrong with this.
late period
• Archives
• W. thinks that we have mathematics because
of the act of archiving. These archives are the
standards based on which we will expand
mathematics.
• What transforms a proof (which at first is only
an experiment) into a picture is the act of
archiving the actual (typed down) proof. This is
how a proof turns into a paradigm.
late period
• Consistency
• Regarding consistency, W. thinks that mathematicians are mistaken in being so
much concerned about the proof of the consistency of axioms. “I have the feeling
that if there were a contradiction in the axioms of a system it wouldn't be such a
great misfortune. Nothing easier than to remove it.”
• The idea that a contradiction is not harmful per se is one Wittgenstein made
repeatedly. In his discussion of Gödel's incompleteness theorems, for instance, he
says: “Is there harm in the contradiction that arises when someone says: ‘I am lying.—
So I am not lying.—So I am lying.—etc.’ ? I mean: does it make our language less
usable if in this case, according to the ordinary rules, a proposition yields its
contradictory, and vice versa?—the proposition itself is unusable, and these
inferences equally.… Such a contradiction is of interest only because it has tormented
people.”
• But, Wittgenstein seems to have been laboring under the misconception that we can
repair a contradictory system simply by refusing to draw any conclusions from a
contradiction. But regarding contradiction he never really developed this material
into a stable account.
conclusion
• What will distinguish the mathematicians of
the future from those of today will really be a
greater sensitivity […] people will then be more
intent on absolute clarity than on the discovery
of new games.
• Philosophical clarity will have the same effect
on the growth of mathematics as sunlight has
on the growth of potato shoots. (In a dark
cellar they grow yards long.)
conclusion
• A mathematician is bound to be horrified by
my mathematical comments, since he has
always been trained to avoid indulging in
thoughts and doubts of the kind I develop. […]
he has acquired a revulsion from them as
infantile. That is to say, I trot out all the
problems that a child learning arithmetic, etc.,
finds difficult […] I say to those repressed
doubts […] demand clarification!
further reading
• Tractatus Logico-Philosophicus (§6.2 - §6.36311)
• Philosophical Investigations (§185 - §238)
• Remarks on the Foundations of Mathematics (§1 - §5)
references
• Glock, Hans-Johann. 1996. A Wittgenstein Dictionary. Blackwell Oxford.
• Addis, Mark. 2006. Wittgenstein: A Guide for the Perplexed. Continuum.
• Potter, Michael. 2011. “Wittgenstein on Mathematics.” In The Oxford
Handbook of Wittgenstein, edited by Oskari Kuusela and Marie McGinn.
Oxford University Press.
• Mácha, Jakub. 2015. Wittgenstein on Internal and External Relations:
Tracing All the Connections. Bloomsbury Publishing.
• Rodych, Victor. 2018. “Wittgenstein’s Philosophy of Mathematics.” In The
Stanford Encyclopedia of Philosophy, edited by Edward N Zalta.
• Bangu, Sorin. 2012. “Ludwig Wittgenstein: Later Philosophy of
Mathematics.” Internet Encyclopedia of Philosophy.
thank you