Apresentação no FÓRUM ABERTO - PPGFIL-UNICAP, Recife, 19/Mar/2021

1. Linguagem, Lógica e a
Natureza da Matemática
Ruy J.G.B. de Queiroz
Centro de Informática da UFPE
FÓRUM ABERTO - PPGFIL-UNICAP, Recife, 19/Mar/2021

2. Logic, Methodology and
Philosophy of Science VI, 1979

3. Dummett–Prawitz
‘meaning via proofs’
• “the meaning of each [logical]
constant is to be given by
specifying, for any sentence in
which that constant is the
main operator, what is to count
as a proof of that sentence, it
being assumed that we
already know what is to count
as a proof of any of the
constituents.” (p. 12, Michael
Dummett in Elements of
Intuitionism (1977))

4. Dummett–Prawitz
‘meaning via proofs’
• “As pointed out by Dummett, this
whole way of arguing with its stress
on communication and the role of
the language of mathematics is
inspired by ideas of Wittgenstein
and is very different from Brouwer’s
rather solipsistic view of
mathematics as a language- less
activity. Nevertheless, as it seems,
it constitutes the best possible
argument for some of Brouwer’s
conclusions. (p. 18, Dag Prawitz
‘Meaning and proofs: on the
con
fl
ict between classical and
intuitionistic logic’, Theoria
(Sweden) XLIII:2–40, 1977.)

5. The 2020 Rolf Schock Prize
in logic and philosophy

6. Linguagem versus
“Realidade”
•
“O mundo é a totalidade
de fatos, não de coisas”
•
“Os limites de minha
linguagem signi
fi
cam os
limites de meu mundo”
•
“Sobre o que não podemos
falar, devemos passar em
silêncio”
Ludwig Wittgenstein (1889-
1951)

7. Lógica: Ciência da Argumentação,
Princípios da Racionalidade
• Aristóteles (380a.C.– 322
a.C.): o que há por trás da
argumentação “racional”
• Leibniz (1646-1716):
linguagem simbólica visando
objetividade
• Boole (1815-1864): “álgebra
do pensamento”
• Frege (1848-1925): linguagem
para escrever conceitos

8. Gottlob Frege:
Lógica de Predicados e λ-Cálculo
• Begriffsschrift (1879)
• Werthverlauf (1891-3): éf(e)
• Frege’s éΦ(e) vs. Church’s
λeΦ(e)

9. “Toda Filoso
fi
a é uma Crítica
da Linguagem”
• “Escreví um livro chamado
Tractatus Logico-
Philosophicus contendo todo
o meu trabalho dos últimos 6
anos. Acredito que resolví
nossos problemas
fi
nalmente.
Isso pode parecer arrogante,
mas não consigo evitar.”
(1919)

10. Filoso
fi
a como
Crítica da Linguagem
• Fritz Mauthner (1849-1923)
• Wittgenstein: “Toda
fi
loso
fi
a é ‘crítica da
linguagem’. (Todavia, não no sentido de
Mauthner)” (Tractatus, 4.0031)
• Linguagem como Spielregel.
• “Linguagem nada mais é do que seu
uso.”
• “Se eu quiser ascender à crítica da
linguagem, que é a mais importante
ocupação da humanidade pensante,
então devo destruir a linguagem por trás
de mim e em mim, passo a passo: tenho
que destruir todos os degraus da
escada enquanto estou escalando-a.”
(B. i. 2)

11. Bertrand Russell (1872-1970) (co-
autor do “Principia Mathematica”)
• “O Tractatus Logico-Philosophicus do
Sr. Wittgenstein, se chega ou não a
revelar a verdade última sobre os
assuntos de que trata, certamente
merece, pela sua amplitude e o alcance
e profundidade, ser considerada um
evento importante no mundo
fi
losó
fi
co.
A partir dos princípios do Simbolismo e
as relações que são necessárias entre
palavras e coisas em qualquer idioma,
o trabalho aplica o resultado desta
busca a vários departamentos da
fi
loso
fi
a tradicional, mostrando em cada
caso como a
fi
loso
fi
a tradicional e as
soluções tradicionais surgem da
ignorância dos princípios do
Simbolismo e como resultado do mau
uso da linguagem.” (Maio, 1922)

12. Família Wittgenstein em
Viena
• Karl Wittgenstein: 2o mais rico do
Império Áustro- Húngaro (atrás dos
Rotschilds). Dono da Siderúrgica Krupp
• Klimt pintou Gretl (irmã)
• Ravel compôs peça (mão
esquerda) para Paul
• Brahms, Mahler e Strauss
freqüentavam a casa
• 1919: renúncia à herança

13. Professor primário, Jardineiro,
Arquiteto (1919-1928)

14. Palestra em Viena (Março/1928):
“Matemática, Ciência e Linguagem”
• Mathematics is not formal. The objects
of mathematics are mental constructions
in the mind of the (ideal) mathematician.
Only the thought construction of the
(idealized) mathematician are exact.
• Mathematics is independent of the
experience in the outside world, and it is
in principle also independent of
language. Communication by language
may serve to suggest thought
constructions to others, but there is no
guarantee that these constructions are
the same.
• Mathematics does not depend on logic;
on the contrary, logic is a part of
mathematics.

15. Luitzen E. J. Brouwer
(1881-1966) e o Intuicionismo
•
Matemática também como um
ato de criação, e não apenas de
descoberta. Objetos matemáticos
como construções mentais.
•
A concepção de espaço não é
a priori, embora a de tempo o
seja.
•
Controvertido, porém
respeitado: contribuiu para a
consolidação da Topologia.
• Na Universidade aos 16.

16. Choice Sequences
• “In intuitionistic mathematics, a choice sequence is a constructive
formulation of a sequence.”
• “A distinction is made between lawless and lawlike sequences.”
• “A lawless (also, free) sequence is one that is not predetermined. It is to
be thought of as a procedure for generating values for the arguments 0, 1,
2, .... That is, a lawless sequence α is a procedure for generating α0,
α1, ... (the elements of the sequence α) such that:
• At any given moment of construction of the sequence α only an initial
segment of the sequence is known, and no restrictions are placed on
the future values of α; and
• One may specify, in advance, an initial segment ⟨α0,α1,…,αk⟩
of α”

17. “The unreliability of the logical principles”
(1908)
“it must be held uncertain whether questions such as:
‘Is there in the decimal expansion of π a digit that
occurs enduringly more often than all others?’
‘Do there occur in the decimal expansion of π
in
fi
nitely many pairs of equal consecutive digits?’
have a solution.”

18. “Mathematical languages shape our
understanding of time in physics”
(Nicolas Gisin, Fev 2020)
• “Mathematics is the language of physics and Platonistic
mathematics makes it dif
fi
cult to talk about time. Hence,
the sense of
fl
ow of time was also expulsed from physics:
all events are the ineluctable consequences of some
‘quantum
fl
uctuations’ that happened at the origin of time
— the Big Bang. Accordingly, in today’s physics there is no
‘creative time’ and no ‘now’.”
• “This had dramatic consequences, in particular when one
remembers that physics is not only about technologies
and abstract theories, but also about stories on the
workings of nature. Time is an indispensable ingredient in
all human narratives.”

19. Wittgenstein e Brouwer
• “On March 10, 1928, Brouwer lectured in Vienna on
his intuitionistic foundations of mathematics. Ludwig
Wittgenstein attended that lecture, persuaded by
Herbert Feigl, who afterwards wrote about the hours
he spent with Wittgenstein and others after the lecture:
a great event took place. Suddenly and very volubly
Wittgenstein began talking philosophy – at great
length. Perhaps this was the turning point, for ever
since that time, 1929, when he moved to Cambridge
University, Wittgenstein was a philosopher again, and
began to exert a tremendous in
fl
uence.”

20. Livro Azul (1930)
• “As perguntas ‘O que é
comprimento?’, ‘O que é
signi
fi
cado?’, ‘O que é o
número 1?’, etc., produzem
em nós uma cãibra mental.
• Sentimos que não podemos
apontar para nada em
resposta a elas, e mesmo
assim deveríamos apontar
para algo.”

21. Santo Agostinho e o
Aprendizado de Linguagem
• “Quando Santo Agostinho fala sobre o aprendizado de
linguagem ele fala sobre como associar nomes a coisas,
ou entender os nomes das coisas.
• Nomear aqui aparece como os fundamentos, a base e
tudo o que diz respeito a linguagem.
• Nessa visão de linguagem encontramos as raízes da
seguinte idéia: Toda palavra tem um signi
fi
cado. Esse
signi
fi
cado está correlacionado com a palavra. É o objeto
para o qual a palavra aponta.”
•

22. Jogos de Linguagem
• “Vou chamar o todo, consistindo de linguagem e as ações
nas quais ela está entrelaçada, de ‘jogo de linguagem’.”
• “A palavra ‘jogo-de-linguagem’ é usada aqui para
enfatizar o fato de que falar numa linguagem é parte de
uma atividade, ou uma forma de vida.”
• “Quando penso através da linguagem, não existem
‘signi
fi
cados’ passando pela minha mente em conjunto
com as expressões verbais: a linguagem é em si mesma o
veículo do pensamento”. (Não há linguagem privada).

23. Signi
fi
cado como
“Condições de Verdade”
• Alfred Teitelbaum/Tarski
(1901-1983) é mais conhecido
por seu trabalho em teoria dos
modelos, metamatemática, e
lógica algébrica.
• Em 1933 publicou artigo sobre
“uma de
fi
nição de ‘verdade’
em linguagens formalizadas”,
usando “condições de
verdade”, via a distinção entre
linguagem e metalinguagem.
Dá suporte teórico à “teoria da
correspondência”.

24. Gramática Filosó
fi
ca (1933)
• “But an interpretation is
something that is given in
signs. It is this interpretation
as opposed to a different one
(running differently).—So
when we wanted to say “Any
sentence still stands in need
of an interpretation”, that
meant: no sentence can be
understood without a rider.”
(§9, p. 47 of Philosophical
Grammar, Part I.)

25. Incompletude
• Kurt Gödel (1906-1978),
fi
cou conhecido
por seu trabalho no uso de lógica e
teoria dos conjuntos para entender os
fundamentos da matemática.
• 1930:
1. “Qualquer formalização da
Aritmética ou é incorreta ou é
incompleta.”
2. “Nenhuma teoria formalizada
pode provar sua própria
consistência.”
3. “Nem toda verdade aritmética é
demonstrável [na teoria formal da
Aritmética].”

26. Investigações Filosó
fi
cas
(1945)
• “But how can a rule shew me
what I have to do at this point?
Whatever I do is, on some
interpretation in accord with
the rule.”—That is not what we
ought to say, but rather: any
interpretation still hangs in the
air along with what it
interprets, and cannot give it
any sup- port. Interpretations
by themselves do not
determine meaning.” (§198,
Part I, p. 80e, Philosophical
Investigations.)

27. Signi
fi
cado como
“Condições de Prova”
• Arend Heyting (1898-1980),
discípulo de Brouwer, é mais
conhecido pela chamada
“interpretação de Brouwer-
Heyting-Kolmogorov” da
linguagem da matemática.
• Em 1930 publica artigo
de
fi
nindo o signi
fi
cado de
proposições matemáticas por
suas “condições de prova”.

28. Brouwer-Heyting-Kolmogorov
• a proof of the proposition consists of
• A & B a proof of A and a proof of B
• A ∨ B a proof of A or a proof of B
• A ⊃ B a method which takes any proof
of A into a proof of B
• (∀x) B(x) a method which takes an arbitrary
individual a into a proof of B(a)
• (∃x) B(x) an individual a and a proof of B(a)

29. Remarks on the Foundations
of Mathematics (1941-4)
• “I once said: ‘If you want to know
what a mathematical proposition
says, look at what its proof
proves’ (Phil. Grammar, p. 369).
Now is there not both truth and
falsehood in this? For is the sense,
the point, of a mathematical
proposition really clear as soon as
we can follow the proof?” (§10, Part
VII, p. 367)
• “The proof of a proposition certainly
does not mention, certainly does not
describe, the whole system of
calculation that stands behind the
proposition and gives it its sense.”
(§11, Part VI, p. 313)

30. Intuicionismo como
Veri
fi
cacionismo
• Michael Dummett (1925-2011),
conhecido por seu trabalho sobre
verdade e signi
fi
cado e suas
implicações para os debates entre
realismo e anti-realismo, um termo
que ele ajudou a popularizar.
• Retomada do Intuicionismo, porém
com base no paradigma
“signi
fi
cado é uso” de Wittgenstein
• O signi
fi
cado de um enunciado
matemático é determinado pelo
que conta como uma prova dele.
(1977)

31. Teoria Veri
fi
cacionista do
Signi
fi
cado
• Dag Prawitz (1936-), mais
conhecido por seu trabalho em
teoria da prova e os fundamentos
da dedução natural.
• Teoria do Signi
fi
cado baseada na
idéia de que o modo de provar
um enunciado determina seu
signi
fi
cado.
• Junta-se a Dummett para
reformular o Intuicionismo sem o
exoticismo de Brouwer
(“Matemática é alingüística”).

32. Veri
fi
cacionismo e
Julgamento
• Per Martin-Löf (1942-)
• “the meaning of a proposition is the
method of its veri
fi
cation.”
• “the explanations of the meanings of
the logical constants, the connectives
and the quanti
fi
ers, given by Brouwer,
Heyting and Kolmogorov: they all follow
the common pattern that, whatever the
logical constant may be, an
explanation is given of what a proof of
a proposition formed by means of that
logical constant looks like, that is, what
is the form, and, more precisely,
canonical or direct form, of a proof of a
proposition which has that speci
fi
c
logical constant as its outermost sign. “

33. Dummett, 1977
• “the meaning of each [logical] constant is to be
given by specifying, for any sentence in which that
constant is the main operator, what is to count as a
proof of that sentence, it being assumed that we
already know what is to count as a proof of any of
the constituents.”

34. Prawitz, 1977
• “As pointed out by Dummett, this whole way of arguing with its
stress on communication and the role of the language of
mathematics is inspired by ideas of Wittgenstein and is very
different from Brouwer's rather solipsistic view of mathematics as a
languageless activity. Nevertheless, as it seems, it constitutes the
best possible argument for some of Brouwer's conclusions.”
• I have furthermore argued that the rejection of the platonistic
theory of meaning depends, in order to be conclusive, on the
development of an adequate theory of meaning along the lines
suggested in the above discussion of the principles concerning
meaning and use. Even if such a Wittgensteinian theory did not
lead to the rejection of classical logic, it would be of great interest
in itself.”

35. Semântica Baseada em
Jogo ou Diálogo
• Paul Lorenzen (1915-1994),
Jaakko Hintikka (1929-2015)
• Signi
fi
cado de uma sentença
na linguagem da matemática
de
fi
nido pela forma de
interação, ao invés de suas
“condições de verdade”.
• Dialogue Semantics
(Lorenzen) vs Game-
Theoretical Semantics
(Hintikka)

36. Dialogue Semantics
(Lorenzen, 1955)
• If the defender X states "A or B", the challenger Y has the right to ask him to choose
between A and B.
• If the defender X states "A and B", the challenger Y has the right to choose between
asking the defender to state A or to state B.
• If the defender X states that "if A then B", the challenger Y has the right to ask for B by
granting herself (the challenger) A.
• If the defender X states "no-A", then the challenger Y has the right to state A (and then
she has the obligation to defend this assertion).
• If the defender X states for "all the x's it is the case that A[x]", the challenger Y has the
right to choose a singular term t and ask the defender to substitute this term for the free
variables in A[x].
• If the defender X states "there is at least one x, for which it is the case that A[x]", the
challenger Y has the right to ask him to choose a singular term and substitute this term
for the free variables in A[x].

37. Game-Theoretical
Semantics (GTS)

38. Signi
fi
cado e
Conseqüências
“Will you think that I have gone mad if I make the fol- lowing suggestion?: The
sign (x).φx is not a com- plete symbol but has meaning only in an inference of
the kind: from ⊢ φx⊃xψx.φ(a) follows ψ(a). Or more generally: from ⊢
(x).φx.ε0(a) follows φ(a). I am—of course—most uncertain about the matter but
something of the sort might really be true.”
Letter R.3 Russell, dated 1.7.12, p. 12 of Wittgenstein, L.: 1974, Letters to
Russell, Keynes and Moore, Ed. with an Introd. by G. H. von Wright, (assisted
by B. F. McGuinness), Basil Blackwell, Oxford, 190pp.
•

39. Signi
fi
cado e
Conseqüências
“That the proposition “φa” can be inferred from the proposition “(x).φx” shews how generality is
present even in the sign “(x).φx”.
(dated 24.11.14, p. 32e
of the Notebooks
)
“before a proposition can have a sense, it must be completely settled what propositions follow from it”.
§3.20103, p. 65 of the Prototractatus
“That the proposition “φa” can be inferred from the proposition “(x).φx” shews how generality is
present even in the sign “(x).φx”.
dated 24.11.14, p. 32e
of the Notebooks
“(The possibility of inference from (x).fx to fa shows that the symbol (x)fx itself has generality in it.)”
Tractatus, §5.1311, p. 38.
•

40. Type Theory vs
Labelled Natural Deduction
• Being an ‘enriched’ system of
natural deduction, it helps to
formulate logical calculi in an
operational manner. By
uncovering a certain harmony
between a functional calculus
on the labels and a logical
calculus on the formulas, it
allows mathematical
foundations for systems of
logic presentation designed to
handle meta-level features at
the object-level via a labelling
mechanism. (2011)

41. Impact of Intuitionistic Type
Theory in Mathematics
• By introducing (in 1973) a framework in which a
formalisation of the logical notion of equality, via the so-
called “identity type”, Martin-Löf’s Type Theory allows
for a surprising connection between term rewriting and
geometric concepts such as path and homotopy.
• The impact in mathematics has been felt more strongly
since the start of Vladimir Voevodsky’s program on the
univalent foundations of mathematics around 2005,
joined by Steve Awodey in building an approach
referred to in 2007 as homotopy type theory.

42. Fundamental Groups
• “In modern mathematics it is common to study a
category by associating to every object of this
category a simpler object that still retains suf
fi
cient
information about the object of interest. Homotopy
groups are such a way of associating groups to
topological spaces.
• That link between topology and groups lets
mathematicians apply insights from group theory to
topology. ” (Wikipedia)

43. A Topological Application of Labelled Natural Deduction
Tiago M. L.Veras, Arthur F. Ramos, Ruy J. G. B. de
Queiroz, Anjolina G. de Oliveira (2020)
• We use a labelled deduction system based on the
concept of computational paths (sequences of
rewrites) as equalities between two terms of the
same type.
• We then proceed to show the main result here:
using this system to obtain the calculation of the
fundamental group of the circle, of the torus and
the real projective plane.