Transcript Diagrams in logic and mathematics

When is a diagram precise and
accurate?
Handmade drawings can be
rather rough, but still
intelligible
Diagrams can be
accompanied by labels and
sentences
When does diagrams
support reasoning?
PROGRAMME
Centro de Filosofia das Ciências da Universidade de Lisboa
January 14-16, 2008
• Workshop 1 (Jan. 14):
Diagrams in logic and mathematics: epistemological questions
• Workshop 2 (Jan. 15):
Visual displays of information and the practice of science: pragmatic
approaches
• Public Conference (Jan. 16)
Diagrammatic reasoning: Empirical research and theoretical questions
Valeria Giardino
Università di Roma ‘La Sapienza’
Institut Jean Nicod
[email protected]
http://valeria.giardino.googlepages.com/
Centro de Filosofia das Ciências da Universidade de Lisboa
January 14, 2008
Workshop1
Diagrams in Logic and
Mathematics
Epistemological Questions
Valeria Giardino
Università di Roma ‘La Sapienza’
Institut Jean Nicod
[email protected]
http://valeria.giardino.googlepages.com/
A very simple (and ‘visualized’) example:
The Pythagorean Theorem
(a+b)2 = c2 + 2ab
Outline
• Introduction: ‘Logocentric’ views of logic and
mathematics
• I. First alternative: Heterogenous Logic
• II. Second alternative: A form of Platonism
• III. Third alternative: Discovery without
justification as a conceptual possibility
• Questions for the discussion
Introduction
The ‘dogma’ of logocentricity
Logocentricity/1
Sheffer (1926)
Review of the second edition of Whitehead and
Russell’s Principia Mathematica
“Throughout the four volumes of Principia the
authors are concerned with one main problem –
namely, the proof of the thesis that pure mathematics
is an extension of formal logic, and is nothing else.”
Dialectical and
psychological
speculation
regarding the
foundation of
mathematics
Actual demonstration
that mathematics either
does, or does not,
involve concepts or
processes than are
extra-formal
With the epistemology of mathematics – the
question of the «objectivity» or «subjectivity» of
mathematical «reality» - and with the psychology of
the act of mathematicizing, Whitehead and Russell
have no concern.
The logical status of mathematics – the question
of the derivability or the non-derivability of
pure mathematics from formal logic alone.
Just as the proof of certain theories in
metaphysics is made difficult, if not hopeless,
because of the «egocentric» predicament, so the
attempt to formulate the foundations of logic is
rendered arduous by a corresponding
«logocentric» predicament. In order to give an
account of logic, we must presuppose and employ logic.
Three basic norms
(a)They are assuming the validity of logic: their aim is not to
validate logic, but only to make explicit, at least in part, that
which we have assumed to be valid.
(b)They must keep the study of formal structure – the
question of notation – entirely distinct from the
investigation of the «loci» or interpretations of a structurecomplex.
(c)Finally, if they are to surmount the difficulties created by
the «logocentric» predicament, they must discriminate
sharply between the problems of «Symbolic Analysis» - i.e.
notation and interpretation - and the study of the
conditions that make the notational and the interpretational
phases significant and valid.
Logocentricity/2
Tennant (1984)
[The diagram] is only an heuristic to prompt certain trains of
inference; [...] it is dispensable as a proof-theoretic device;
indeed, [...] it has no proper place in the proof as such. For
the proof is a syntactic object consisting only of sentences
arranged in a finite and inspectable way.”
A shift from a problem about foundations to a ‘dogma’ to an
issue which concerns what a proof is
...what about the epistemology of mathematics?
I.
First alternative:
Heterogeneous
Logic
Barwise and Etchemendy’s project
(1996)
“[...] visual forms of representation can be
important, not just as heuristic and pedagogical
tools, but as legitimate elements of mathematical
proofs. As logicians, we recognize that this is a
heretical claim, running counter to centuries of
logical and mathematical tradition”
Against the dogma: why?
Shin (2004)’s reconstruction
1. Students' performances: understanding
semantic concepts can be of help for carrying out formal
proofs in a deductive system
2. Reasoning is a heterogeneous activity
people use multiple representations of information while
reasoning, and those representations are often in nonsentential forms such as diagrams
3. Unity of teaching and reasearch
the practical power of multi-modal reasoning together with
modern logic and its formalization and rigour
 expanding the territory of logic by freeing it
from a mode of representation only
First problem
“All of us engage in and make use of valid
reasoning, and in the process of reasoning
human beings obtain information through many
different kinds of media, including diagrams,
maps, smells, sounds, as well as written or
spoken statements”
Shin (2004)
Second problem
“to do with visual/spatial reasoning something analogous
to what Frege and his followers did for the
formal/linguistic one”  towards a new standardized
system
Axioms to account for diagrams and for our way of
referring to them in reasoning?
“[…] the work carried out by Barwise and Etchemendy on
visual arguments in logic and mathematics is motivated
in great part by the proof-theoretic foundational
tradition”
Mancosu (2005)
II. Second alternative:
A form of Platonism
Brown (1999)
“Some ‘pictures’ are not really pictures, but rather are
windows to Plato’s heaven”
It is possible to have a realist view of mathematics
and its objects, without being at the same time a
realist view of pictures
“As telescope help the unaided eye, so some
diagrams are instruments (rather than
representations) which help the unaided mind’s
eye”
Problems
1. ontological statements are in general much more
problematic than the difficulties they are intended to
find solution to
the infinite debate about the ontological nature of
mathematical objects shows that it is very problematic
to settle ontological questions once and for all
Balaguer (1998)
2. even if we accept that ontology can represent some
methodological advantages, this is not always the case
depending on the problem in question, it could be more
useful to think about the object of research
differentely
III. Third
alternative:
Discovery (by
visualizing)
without
justification
First...
Something on discovery and
mathematics
Hadamard (1949)
Mathematical discovery – or invention – involves
first a preparation and an illumination which are
mostly subconscious; then, this illumination is
verified and concluded by means of conscious
thinking, which is nevertheless necessary in
teaching mathematics.
Polya (1945)
How do we solve it?
- Understanding the plan
- Devising the plan
- Carrying out the plan
- Looking back
Exposition “which progresses relentlessly from the
data to the unknown and from the hypothesis to
the conclusion”:
perfect for checking the argument in detail BUT far
from being perfect for making understandable the
main line of the argument!
“A detail pictured in our imagination may be
forgotten; but the detail traced on paper
remains, and, when we come back to it, it
reminds us of our previous remarks, it saves us
some of the trouble we have in recollecting our
previous consideration.”
Rota (1997)
“the laws of logic are not sculpted in stone, eternal and
immutable. A realistic look at the development of
mathematics shows that the reasons for a theorem are
found only after digging deep and focusing upon the
possibility of a theorem. The discovery of such hidden
reasons is the work of the mathematician. Once such reasons
are found, the choice of particular formal statements
which express them is secondary. Different but
exchangeable formal versions of the same reason can
and will be given depending on the circumstances”
“this pretence of ‘identifying’ mathematics with a style of
exposition is having a corrosive effect on the way
mathematics is viewed by scientists in other disciplines”
Giaquinto’s approach
Epistemological (evaluative)
questions to ask
• Can a visual way of acquiring a mathematical
belief justify our believing in it?
• Can a visually way of acquiring mathematical
belief be knowledge in the absence of
indipendent non-visual grounds?
• What level of confindence would be rational?
Conditions for mathematical discovery:
- To believe something independentely
coming to see it by one’s own lights, rather than by
reading it or by being told
- To believe something in a epistemically acceptable
way
a way of acquiring a belief is epistemically
acceptable if it is reliable as a way of getting
belief and it involves no violation of epistemic
rationality in the circumstances
- Absence of irrationality in getting a belief does
not entail that the believer has a justification
- Justification is not a further conditions of
discovery
Discovery without justification is at least a
conceptual possibility; in mathematics, this
possibility is quite often realized
Proofs often post-date discovery
 Back to an epistemology of individual
discovery
Questions for the discussion
I.
Is an epistemology of mathematics supposed
to discuss the role of visualization in
mathematics?
II. Is it necessary to consider visualization in the
proofs or as an “underlying mental
operation”?
III. What is the difference between these two
approaches?
IV. Mental operations or external supports?
References
Barwise - Etchemendy (1996), ‘Visual Information and Valid Reasoning’, in Allwein Barwise (eds.) (1996), Logical Reasoning with Diagrams, 3-25.
Brown (1999), Philosophy of Mathematics: an introduction to the world of proofs and pictures.
Giaquinto (2007), Visual Thinking in Mathematics.
Polya (1945), How to solve it.
Rota (1997), ‘The Phenomenology of Mathematical Proof ’, in Kamamori (ed.), Proof and
Progress in Mathematics, 183-196.
Mancosu (2005), ‘Visualization in Logic and Mathematics’, in Mancosu et al. (eds.) (2005),
Visualization, Explanation and Reasoning Styles in Mathematics, 13-30.
Sheffer (1926), ‘Review of A. N. Whitehead and B. Russell, Principia Mathematica’, in Isis
(8), 226-231.
Shin (2004), ‘Heterogenous Reasoning and its Logic’, in The Bullettin of Symbolic Logic (10)
86-106.
Tennant (1984), The withering away of formal semantics.