Philosophy Lecture

1. Mathematical Argumentation and the Toulmin Model
Brendan Larvor
www.herts.ac.uk/computerscience

2. Toulmin: The Uses of Argument
1958
• Data-Warrant-Conclusion
model
(based on legal
argumentation)
• Field-Specific Modalities and
Inferences

3. The D-W-C model of inference
One among many

4. Field-Specific Modalities and Inferences
You can’t do that here!

5. Mathematical Fields
In search of non-arbitrary individuation
Toulmin is not much help. He doubts,
“…whether any genuine, practical argument could ever be
properly analytic.
Mathematical arguments alone seem entirely safe:…This unique
character of mathematical arguments is significant. Pure
mathematics is possibly the only intellectual activity whose
problems and solutions are ‘above time’. A mathematical
problem is not a quandary; its solution has no time-limit; it
involves no steps of substance.”
(Uses of Argument p.127)

6. Mathematical Fields
In search of non-arbitrary individuation
Candidates for field-specific argumentation
Euclid – see Ken Manders’ work on the Euclidean
diagram
Galileo – see Grosholz’s analysis of his use of
controlled ambiguity in the elements of diagrams
But perhaps these are not pure enough to count as
counterexamples to Toulmin’s claim.
Similarly for visualisation (see Giaquinto)

7. Mathematical Fields
In search of non-arbitrary individuation
In any case, mathematics now works by mapping
between fields
Early algebra and geometry (complex numbers needed
to calculate cubics by radicals)
Analytic proofs of quadratic reciprocity
Prime number theorem: for any real x, the number of
primes <x is asymptotic to x/ln(x)
This was proved first using complex analysis

8. Mathematical Fields
In search of non-arbitrary individuation
So should we give up on mathematical fields?
Future mathematical advances might collapse any pair
of fields into one (i.e. unify their inferential practices)
No: call the historians!

9. Mathematical Fields
In search of non-arbitrary individuation
Van Brummelen: what are the boundaries of
trigonometry?
Is Euclid II.13 trigonometry?
This looks to us now like the law of cosines.
But only if we add the cosine to it!

10. Mathematical Fields
In search of non-arbitrary individuation
BSHM Bulletin 25 (2010) p. 3
Euclid shows:
B2
=c2
+a2
-2a(BD)
Maybe this should count as
trigonometry because it involves
angle measurement?
No! So does the Rhind Papyrus!

11. Mathematical Fields
In search of non-arbitrary individuation
Van Brummelen’s answer:
The Rhind Papyrus is not part of the history of
trigonometry because, as a matter of contingent
historical fact, “Its triangle problems had no effect
(direct or indirect) on any later trigonometric
developments…” (p. 4)
These developments were triggered by “the merging
of Babylonian-style calculation with the geometric
models in Hipparchus’s mathematical astronomy
around 130 BC.”

12. Mathematical Fields
In search of non-arbitrary individuation
Van Brummelen’s answer:
“We have asserted trigonometry to be a kind of
mathematical activity rather than a body of
theorems. Although Euclid has the theorem, he
does not engage in the activity.”
BL: never mind that he might have done!

13. Mathematical Fields
In search of non-arbitrary individuation
Moral: the identity of a field of mathematical practice is
a contingent historical fact, not a mathematical
necessity.
Consequence: field-specific inferences cannot depend
for their intelligibility on field-specific modalities.