Kant and Hume on Mathematical knowledge
(And why Kant was right!)
(INCOMPLETE DRAFT: Liable to change)
Aaron Sloman
http://www.cs.bham.ac.uk/~axs/
School of Computer Science, University of Birmingham, UK
(Philosopher in a Computer Science department)
Installed: 6 Oct 2022
Last updated: 16 Oct 2022
This is
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/kant-hume-mathematics.html
A PDF version may be added later.
This is still an incomplete draft
(Likely to be revised.)
An older, longer document with overlapping contents is
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/kant-maths.html
A partial index of discussion notes in this directory is in
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/AREADME.html
Introduction
The main function of this document is to present some historical, philosophical and cognitive background regarding ancient forms of mathematical discovery, which can be referenced in other documents on this web site, without the contents having to be duplicated.
A 20th Century Educational disaster
Owing to seriously flawed changes in mathematical education from about the
middle of the 20th century, many people, including philosophers, psychologists,
neuroscientists, AI researchers, roboticists, education theorists and
mathematics teachers fail to grasp some important features of mathematical
discovery and mathematical reasoning that were noticed by Immanuel Kant.
Since completion of my 1962 D.Phil thesis, defending Immanuel Kant's philosophy
of mathematics, as presented in his Critique of Pure Reason in *1781, I
have been attempting to explain how a collection of apparently unrelated topics
in philosophy of mathematics, philosophy of knowledge, theories of biological
evolution and development and philosophy of mind have deep connections that are
generally ignored by scientists and philosophers, but are closely related to
Kant's philosophy of mathematics.
*
See:
https://www.gutenberg.org/files/4280/4280-h/4280-h.htm
I now believe the key abilities to make mathematical discoveries, used by ancient mathematicians centuries before well known ancient mathematicians such as Pythagoras, Euclid and Archimedes were born, cannot be explained using current theories in psychology, neuroscience, AI, or philosophy.
That's because the key mechanisms involve chemical structures and their interactions, rather than forms of reasoning or learning proposed in those disciplines.
David Hume
Immanuel Kant
Born 1711
Born 1724
This is an introduction to some of the differences between David Hume's and Immanuel Kant's views on the nature of mathematical knowledge. I have tried to give an account of Hume's views that Kant thought were erroneous, though it is possible that he misunderstood Hume and therefore his criticisms were misdirected.
I apologise for any factual inaccuracy. My main aim is to present Kant's views and explain why they are closer to truth than most views on mathematical knowledge held by philosophers, psychologists, neuroscientists, AI researchers, and others I have encountered. In particular here and in documents referred to below I've tried to defend Kant against his 20th century critics.
Hume's views are not the main topic -- I mention them only because Kant claimed that reading Hume woke him from his "dogmatic slumbers" and caused him to present a rival (non Humean) view of mathematical knowledge as synthetic, not analytic, because discovering many interesting mathematical truths cannot be based simply on the use of logical reasoning applied to definitions of the concepts involved, i.e. mathematical knowledge is synthetic, not analytic. He claimed that mathematical knowledge was also apriori, not empirical, and the contents are necessarily true, not contingent.
EXAMPLE: The triangle sum theorem. Standard Proofs vs Mary Pardoe's rotating arrow proof.
For a long time I have been drawing attention to relationships between ancient human abilities to make discoveries in geometry and topology, and forms of intelligence observable in many non-human species, including nest-building birds, squirrels, and many different primates, including monkeys, bonobos, orangutans, gorillas, etc. (Some of whom have been studied by colleagues in Biosciences in the University of Birmingham -- Jackie Chappell and Susannah Thorpe.)
I did not notice the relevance of hatching processes in eggs of vertebrate species until late 2020. The ideas have been extended several times since then, most recently during 2022, with new links between evolution, in-egg development, and multiple layers of control of species-specific hatching processes, related to species-specific evolutionary history.
I do not claim that these mechanisms can be replicated in AI systems or modelled accurately on computers, insofar as the hatching mechanisms inherently make use of a very powerful mixture of discrete and continuous molecular processes that I suspect cannot be modelled even approximately on digital computers. This is a key difference between chemical processes and processes in digital electronic systems.
The power and generality of chemistry-based biological hatching mechanisms is demonstrated by the huge variety of physical/physiological structures and species specific post-hatching behavioural competences produced by in-egg hatching processes in different species.
Some species that hatch in a relatively under-developed and helpless state, e.g. many bird species that hatch in nests dangerously far from the ground, and have to be fed for some time by their parents before they are ready to fly, at which stage they don't have to learn to fly by training neural networks in trial-and-error processes: that would usually be fatal. Those species will not be discussed only because their mechanisms are more complex than required for the purposes of this document.
Likewise mammals that interact continuously with the mother during pre-natal development, while being fed through an umbilical cord will not be discussed.
There are many other cases that are relevant but not required for the purposes of this document, such as metamorphosis processes in insects, summarised in https://en.wikipedia.org/wiki/Metamorphosis.
Statistics-based neural networks cannot explain intelligence produced in eggs
There are very many researchers in a variety of disciplines trying to use
"trainable neural net" models based on collection of statistical data and
derivation of probabilities, to explain various aspects of intelligence,
apparently unaware that insofar as neural nets collect statistical evidence,
from which they derive probabilities they are incapable of explaining forms of
cognition in which necessary truths, or
impossibilities, are discovered, e.g. the
impossibility of trisecting an arbitrary 2D planar angle using only straight
edge and compasses. Impossibility is not an extremely low probability.
Note:
Ancient human mathematicians discovered a form of geometric construction, known
as the "neusis" construction, which allows an arbitrary angle to be trisected.
Division of the angle into three equal parts is a necessary consequence of use
of the construction: it is not merely a highly probable consequence. So no
statistics based neural net mechanism can establish that neusis works: the
standard proof uses a form of diagrammatic reasoning that extends the forms of
diagrammatic reasoning "axiomatised" by Euclid.
For more details see
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/trisect.html
TO BE CONTINUED
REFERENCES AND LINKS
(To be added)
Discussion of hatching processes in eggs, relevant to Kant's ideas:
https://www.cs.bham.ac.uk/research/projects/cogaff/misc/evo-devo.html
Also see the references in
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/kant-maths.html
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Aaron Sloman
School of Computer Science
The University of Birmingham