By Catarina Dutilh Novaes
This is the final post in my series on beauty, function, and explanation in mathematical proofs (Part I is here; Part II is here; Part III is here; Part IV is here; Part V is here; Part VI is here; Part VII is here). Here I tease out some empirical predictions of the account developed in the previous posts, according to which beauty and explanatoriness will largely (though not entirely) coincide in mathematical proofs. I also comment on how the account, based on a dialogical conception of mathematical proofs, could be made more palatable for those who would prefer a non-relative, absolute analysis of beauty and explanatoriness.
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To summarize, the present account defends the thesis that
when mathematicians employ aesthetic vocabulary to describe proofs, both
positively (‘beautiful’, ‘elegant’) and negatively (‘ugly’, ‘clumsy’), they are
by and large (though not exclusively) tracking the epistemic property of
explanatoriness (or lack thereof) of a proof. Up to this point, the account is
compatible with both subjective (agent-relative) and objective understandings
of beauty and explanation, so long as the two dimensions go together (i.e. both
understood as either subjective or as objective). However, on the basis of a
dialogical conception of mathematical proofs, I’ve also argued that both
explanation and beauty are essentially relative notions with respect to proofs:
an explanation is not explanatory an sich,
but rather explanatory for its intended audience; and if a proof is deemed
beautiful to the extent that it fulfills this explanatory function, then beauty
too emerges as a relative notion.
I’ve also suggested ways in which the present account can be
made more palatable for those who strongly prefer objective accounts of
explanatoriness and beauty. By maximally expanding the range of Skeptics who
will deem a proof explanatory – and so aiming towards the notion of a universal audience – in the limit
(idealized) case a proof may be deemed explanatory by all (i.e. those who have
the required expertise to understand it in the first place). On this conception
then, a proof may also be understood to be beautiful in an absolute sense, i.e.
insofar it fulfills its explanatory function towards any potential (suitably
qualified) audience. The conception of beauty
as fit defended by Raman-Sundström (2012), which relies on an objectively
conceived notion of fit,[1] may
be viewed as an example of such an account, and indeed her description of fit
bears a number of similarities with concepts typically associated with
explanatoriness.[2]
Besides the objective (absolute) vs. subjective (relative) divide,
we’ve also discussed the reductive vs. literal divide. Now, while with respect
to the former I take a clear stance favoring a subjective, agent-relative
account of explanation and beauty, with respect to the latter the present
proposal defends a conciliatory position through the key concept of functional
beauty. If the primary function of a proof is to produce explanatory persuasion
in a suitably equipped audience, and if beauty arises (among other reasons)
from something performing its function well, then a proof will be beautiful to
the extent that it performs its epistemic function(s) well. And thus, we can
say that mathematicians are both tracking epistemic properties and truly making aesthetic judgments at
the same time when using aesthetic vocabulary to describe proofs. But I’ve also
suggested that there may well remain a residue of non-epistemic beauty, for
example in the experience of beauty that may emerge with a proof that proceeds
in unexpected ways.
At any rate, explanation itself seems to be a
multi-dimensional notion (as also suggested by Hafner and Mancosu (2005)), and
a proof may be deemed explanatory for a variety of different reasons: conceptual
clarification, generalization or unification, purity,[3] computational
effectiveness etc. And thus, via a multi-faceted conception of explanation, we
arrive at a multi-faceted conception of the function(s) of a mathematical
proof, which in turn yields a multi-faceted conception of the beauty (or
ugliness) of a proof. This is a desirable result in view of the pluralistic,
non-monolithic conception of mathematical proof that emerges from careful
examination of the varieties of mathematical practices (in particular, with
respect to proofs) – what Wittgenstein in the Remarks on the Foundations of Mathematics (III-46A, III-48)
describes as the ‘motley’ of mathematical techniques of proof.
The present account also gives rise to fairly precise
empirical predictions, which means that it can be put to empirical test,
perhaps along the lines of the work of Inglis and Aberdein (for some of these
predictions, at least some preliminary results already exist):
- Beauty and explanatoriness
are used as comparative, graded notions. The claim is that these are
notions predominantly used to express predilections for certain proofs over
others, and thus are essentially comparative. The results by Inglis and
Aberdein seem to support this interpretation of these notions.
- Mathematicians
disagree in their assessments of the explanatoriness of proofs. This
follows straightforwardly from the claim that the explanatoriness of a proof is
audience-relative. Colyvan (2012, 79) suggests (possibly based on personal
observation) that there is indeed quite some disagreement among mathematicians
on which proofs are explanatory and which are not (or less). Inglis and
Aberdein’s (ms.) results on diversity in proof appraisal suggest this much, but
the questions posed in their study did not address explanatoriness
specifically. So further research on this specific issue is required.
- Mathematicians
disagree in their aesthetic judgments about proofs. Again, this follows
from the claim that explanatoriness (and thus beauty) is an audience-relative
notion, though it may well be that certain kinds of proofs (e.g. ‘brute force’
combinatorial proofs) are unanimously seen as ‘ugly’.[4] Inglis
and Aberdein’s results on diversity in proof appraisal also suggest this much,
but hitherto they only worked with a small range of proofs. We need further
investigation of mathematicians’ appraisals of a much larger range of proofs to
be able to draw more general conclusions on how systematic the phenomenon is.
- Judgments of functionality
and explanatoriness will strongly correlate. This follows from the idea
that the primary function of a proof is to produce explanatory persuasion.
Inglis and Aberdein (2015) found a reasonably strong correlation between
‘explanatory’ and the utility dimension, though less strong than between
‘explanatory’ and the precision dimension, and the negative correlation between
‘explanatory’ and intricacy. However, given that utility and function are not
exactly equivalent notions (utility may also be understood in the sense of the
practical applicability of the proof), here too further work seems to be
required.
- Judgments of beauty
and explanatoriness will correlate, but not coincide. This follows from the
idea that the beauty (or ugliness) of a proof can be largely but not entirely
understood in epistemic terms related to explanatoriness. In the dataset used
in (Inglis and Aberdein 2015), the correlation between beauty and
explanatoriness is greater than zero, but it is not that high (admittedly, lower
than the present account would predict). Here too we might need further
research, for example with a more graded scale of appraisal, to establish the
extent to which mathematicians tend to view what they consider to be
explanatory proofs as also beautiful, or whether a proof can be viewed as
beautiful without being viewed as explanatory, or yet whether two proofs may be
deemed equally explanatory but not equally beautiful, which would suggest that
there is more to beauty than explanatoriness.
- Judgments of beauty
and persuasiveness will correlate. This follows from the claim that to
provoke persuasion is one of the main functions of a proof, and that the beauty
of a proof can be largely understood in functional terms. Indeed, some of the
proofs that are typically viewed as ‘ugly’ such as probabilistic or
computer-assisted proofs tend to be viewed as unpersuasive as well, perhaps
because they fail to transfer to the audience the reason(s) why the theorem
must be true (Easwaran 2009). Inglis and Aberdein (2015) did not include
‘persuasive’ among the adjectives on their list, so at this point a systematic investigation
of the relations between attributions of beauty and attributions of persuasiveness
with respect to proofs remains to be done.
Regarding convergence or dissent in proof appraisal on the
aesthetic and on the explanatory dimensions, the crucial prediction of the
present account is that they will display the same pattern: if mathematicians
largely (dis)agree in their judgments on the explanatoriness of proofs, then
they should largely (dis)agree in their judgments on the aesthetic properties
of proofs, and vice-versa. If this prediction is not corroborated, then this
will constitute a major blow for the present account (which means, in good old
Popperian terms, that the account is falsifiable). At any rate, the fact that
the philosophical analysis presented here gives rise to a number of fairly
precise empirical predictions is, I take it, overall an advantage of the
proposal.
FULL SERIES:
References
Easwaran, Kenny (2009). Probabilistic proofs and transferability. Philosophia Mathematica 17 (3):341-362.
Detlefsen, Michael & Arana, Andrew (2011). Purity of Methods. Philosophers' Imprint 11 (2).
Hafner, Johannes & Mancosu, Paolo (2005). The Varieties of Mathematical Explanation. In Visualization, Explanation and Reasoning Styles in Mathematics. Edited by Paolo Mancosu, et al., 215–250. Berlin: Springer.
Hafner, Johannes & Mancosu, Paolo (2005). The Varieties of Mathematical Explanation. In Visualization, Explanation and Reasoning Styles in Mathematics. Edited by Paolo Mancosu, et al., 215–250. Berlin: Springer.
Inglis, Matthew & Aberdein, Andrew (2015). Beauty Is Not Simplicity: An Analysis of Mathematicians' Proof Appraisals. Philosophia Mathematica 23 (1):87-109.
Inglis, Matthew & Aberdein, Andrew (ms.). Diversity in Proof Appraisal. Available at https://www.academia.edu/8488060/Diversity_in_Proof_Appraisal
Raman-Sundström, Manya (2012). Beauty as Fit: An Empirical Study of Mathematical Proofs.
Proceedings of the British Society for Research into Learning Mathematics 32(3):156-160. Available at http://www.bsrlm.org.uk/IPs/ip32-3/BSRLM-IP-32-3-27.pdf
Raman-Sundström, Manya & Öhman, Lars-Daniel (ms.). Mathematical fit: a case study.
Wittgenstein, Ludwig (1978). Remarks on the Foundations of Mathematics. B. Blackwell.
[1] “Making a distinction between
whether a proof is beautiful and whether a person can grasp that beauty can
help explain phenomena such as why mathematicians judge different proofs to be
beautiful, or why mathematicians and non-mathematicians do the same, without
drawing a necessary conclusion that mathematical beauty is subjective.
Moreover, the metaphor of ‘fit’ suggests a more objective view of beauty might
be warranted— whether a proof is appreciated as beautiful is a subjective
claim, but whether a proof fits a theorem, which relies more on the nature of
the proof than our perception of it, is a more objective one.” (Raman-Sundström
2012, 160)
[2] She presents two kinds of fit:
internal fit as the idea “that the proof directly illuminates what the theorem
is about, providing a sense of why
the theorem is true” (emphasis added); and external fit as the idea “that the
proof derives its beauty from the way it is connected to a family of other
theorems”. (Raman-Sundström 2012, 159) In both cases, the connections with the
notion of explanatoriness as described in the literature are evident. In recent
work (Sundström & Öhman ms.), Sundström further explores the connections
between fit, beauty, and explanation.
[3] A proof is said to be pure if it
only relies on concepts already contained in the statement of the theorem, i.e.
when it does not make use of concepts alien to the theorem itself. See (Detlefsen
and Arana 2011).
[4] Similarly, and as noted by Raman-Sundström
(2012), the fact that mathematicians do not converge in their attributions of
beauty to proofs still does not warrant the conclusion that ‘beauty is (merely)
in the eyes of the beholder’, as there may be a gap “between whether a proof is
beautiful and whether a person can grasp that beauty”. In other words,
disagreement on this dimension would not be sufficient to confirm a subjective
understanding of the beauty of proofs, though convergence on this dimension may
well be sufficient to disconfirm it.
I've tried to think about how your empirical predictions match up with my own impressions of the world I move about in. For some of them it isn't easy, and I think there is nothing for it but to do serious experiments along the lines you suggest. One of the predictions -- that mathematicians disagree in their aesthetic judgments -- is an interesting one, because it seems to me, as you suggest at one point, that there are certain points on which there is almost universal agreement, but there does still seem to be room for disagreement. While I don't have a good concrete example to hand, I do at least have a class of examples. There are some mathematical statements that have proofs that are very short and straightforward once you are fully versed in some powerful machinery that may take a lot of effort to learn, before then becoming thoroughly internalized. To a mathematician who has internalized it, such proofs will seem highly explanatory -- the statement has a clear place in a well-established theoretical framework. But a mathematician who prefers elementary arguments to big-machinery arguments will feel the weight of all the theory that lies behind the "simple" proof and may well be very happy to find an argument that is more complicated but that has far fewer prerequisites. I think there are genuinely situations where there are two proofs with these characteristics, and mathematicians will have strong but different views about which one tells them what is "really going on".
ReplyDeleteWell, that's music to my ears of course :) The example you give illustrates precisely the audience-relativity of the notion of explanatoriness (and beauty) when it comes to proofs.
DeleteBut ultimately most of this should be investigated rigorously rather than being settled on mathematicians' (or even worse, philosophers'!) hunches. Fortunately Inglis and Aberdein are on board with the idea of testing my hypotheses, so there might be some more concrete results in the near future.
Packing "computer-assisted proofs" all in the same box (the box containing "proofs that are typically viewed as ‘ugly’") might not be a good idea. There are nowadays remarkably different categories of computer-assisted proofs in the market: the well-known ones that proceed by exhaustive checking, those that are based on model generation, those based on proof generation, those that are just meant to allow for formal verification, those that take advantage of off-the-shelf decision procedures. These categories may not all have identical measure on a given (stereotypical? ideal?) mathematician's aesthetic ruler.
ReplyDeleteBy the way, here you can find some very good reading on "formal proofs".
I read your final series post and its give us good idea how to solve math equation with proofs and how to find equation answer with proofs thanks for sharing online article summarizer .
ReplyDeleteYou don't overcome challenges by making them smaller but by making yourself bigger
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