Physical/evolutionary foundations for mathematics vs logico/semantic foundations for
mathematics
Aaron Sloman  School of Computer Science
http://www.cs.bham.ac.uk/~axs
_________________________
Notes for an informal seminar in the School of Computer Science, University of
Birmingham Friday 11th September 2016
See also:
http://talks.bham.ac.uk/talk/index/2305
Wittgenstein in (Remarks on the foundations of mathematics):
"For mathematics is after all an anthropological phenomenon."
Does anyone here agree with him?
Who thinks the study of mathematics is the study of products of human minds?
What alternative could there be?
Would that imply that in order to understand the nature of mathematics, we need
to understand human minds?
I suggest it's the other way round:
Understanding, modelling, or replicating the
functions and mechanisms of minds  human and nonhuman, including elephants,
squirrels, crows, dolphins, apes, ... etc. ... requires understanding the nature
of mathematics.
In particular, any theory of consciousness that does not account for varieties
of mathematical consciousness is at least incomplete, and probably wrong.
Mathematical abilities are more widely used by humans and other organisms than
mathematicians and nonmathematicians realise.
The abilities are "layered", like many products of evolution.
AI (including robotics) is currently seriously limited by lack of a theory
explaining widespread, largely unnoticed, mathematical abilities.
Successes like Deep Blue, AlphaGo, impressive robots, speech understanders, all
focus on restricted subsets of intelligent capabilities.
AI lacks a sufficiently general theoretical framework for studying and
replicating intelligence.
Likewise psychology/cognitive science, and philosophy.
E.g. No robotic Euclid is on the horizon,
as far as I know.
Nor a robot with abilities of a human toddler, or a squirrel, or a nestbuilding
bird.
How can we make progress?
Making progress requires us to understand some of the mathematical properties of
our universe, and how that influenced biological evolution.
Including even the very simplest organisms, with the simplest minds
 microbes.
And preverbal human toddlers acquiring and using theories about 3D topology.
Today I want to focus on a very big picture skipping many details.
BIG BANG
...>...
galaxies
...>...
stars
...>...
planets
...>...
organisms
Feel free to email me with alternative answers.
_______________________________________________________
I'll offer a (possibly) new
way of looking at some important subsets of mathematics.
A different analysis may be required for other subsets.
There will not be enough time to cover everything today
 and my own knowledge isn't up to it.
I'll offer a partial answer to the question:
How did the physical universe, in combination with biological evolution,
produce
 structures and processes with mathematical properties,
 mechanisms, e.g. biological control mechanisms, using some of those properties
 individual organisms capable of discovering, reasoning about, and making use
of some of those properties.
 some of whom developed metacognitive abilities to notice their reasoning
processes, think about why they work, communicate them to others, argue about
them, ...
The answer makes use of some observations about construction kits and
construction materials.
Consider these construction kits/materials:

Basic Lego (with bricks only)

Tinkertoys

Basic Meccano

Plasticine

Sand

Foldable paper
Each type has mathematical properties that enable some constructions and rule
out others. E.g. Basic Lego does not allow construction of flexible structures
(unless pieces are assembled in "nonstandard" ways).
Meccano's use of regular spacing between holes makes some shapes impossible to
construct: e.g. not all shapes of triangle can be made using three pieces.
Some construction kits (e.g. basic Meccano)
always require at least two more items to
be added to an existing construction.
Invent your own examples!
Symbolic logic, algebra, set theory all provide construction kits of various
sorts, e.g.
 for constructing formulas
 for constructing proofs
 for constructing searches for proofs or refutations
These can be construed as "abstract" construction kits, with associated
"concrete" construction kits using particular notations.
Levels of abstractness  concreteness.
Fundamental and derived construction kits
Physics and chemistry
The physical universe provides a chemical construction kit, without which living
things found on earth could not exist.
That's because of requirements for
The physical universe also produces lots of nonliving structures and processes
that influence possibilities for life forms (positively or negatively)  e.g.
types of inorganic material, volcanoes, earthquakes, tidal phenomena, on Earth.
All organisms, even the very simplest organisms require abilities to initiate,
control and terminate processes, inside themselves and in the environment.
This requires their
material/physical
resources to be supplemented with
informational
resources.
That includes mechanisms that can use information to determine when to initiate
or terminate various processes, and how to modify ongoing processes, e.g.
speeding up, slowing down, changing direction, or switching from pushing to
twisting, pulling, or bending, etc.
So a universe supporting life must include construction kits for creating
informationprocessing mechanisms.
If the kits available are powerful enough, the processes/mechanisms of natural
selection will use those construction kits to produce increasingly complex
information processing mechanisms
(An information processing arms race.)
We need a detailed study of varieties of types of information processing, the
varieties of construction kit, and the varieties of use of information
processing, e.g. in

reproduction

growth

evolution of new forms

using information about information
Questions about the nature of mathematics
And how it relates to the rest of reality
Questions about the nature of mathematics, the nature of mathematical
discovery, the nature of mathematical proof, how humans can make
mathematical discoveries and how mathematics can be applicable to a
physical world have been raised in the past by many philosophers,
scientists, and mathematicians.
Some examples are summarised here:
https://en.wikipedia.org/wiki/The_Unreasonable_Effectiveness_of_Mathematics_in_the_Natural_Sciences
A related, very old, thread of human history has been concerned with
attempts to create machines that can perform mathematical calculations, and
more recently machines that can find proofs, mechanising processes that had
previously been performed by humans. There are now machines that will find
proofs of new theorems.
One of them will sell you a certified new theorem for
£15.00
http://theorymine.co.uk/
(Nothing in AI so far can replicate the discoveries of Euclid, Archimedes, etc.
or a preverbal human toddler exploring 3D topology.)
(
I have several online papers on limitations of current AI technology, e.g.
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/impossible.html
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/trisect.html
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/triangle.html
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/torus.html
)
Foundational questions  within mathematics
Among many questions still under investigation is whether there is a core
subset of mathematics from which all of mathematics can be rigorously derived:
e.g. some version of symbolic logic, or logic with set theory added. The search
for such a subset is often referred to as the study of “Foundations of
mathematics”, to which great philosophers and mathematicians have
contributed:
https://en.wikipedia.org/wiki/Foundations_of_mathematics
(A possible answer is that no matter how powerful any proposed generative
system is, there are always questions that it cannot answer without first
being extended. Perhaps that also applies to every human brain. Does it
also apply to possible future evolved human brains?)
Epistemological/biological questions
A related but different question is how it came about that humans
could
make mathematical discoveries, including the great discoveries reported
over 2000 years ago by Euclid, Archimedes, Pythagoras, and others, some of
which are still in daily use by engineers and scientists all round the
planet.
What features of human minds or of human modes of reasoning enabled them to
discover mathematical truths, that are neither trivially true, like definitional
truths and their consequences, nor empirical, i.e. substantiated only by
evidence that could be undermind by counterevidence, nor contingent, e.g. true
only because of features of the environment.
E.g. it turned out last century that the question whether physical space is
Euclidean (accurately characterised by Euclid's theories) is empirical, and the
answer is No. But that (so far) applies only to a restricted feature of
Euclidean geometry: the parallel postulate. It can be replaced by alternatives
that are found to fit some regions of physical space better than the parallel
postulate.
A metaphysical foundation for mathematics
Physics/Chemistry and biological evolution by natural selection.
This talk generalises that question: long before there were any human
mathematicians natural selection had produced organisms with mechanisms
that (unwittingly) made use of mathematical structures and processes, e.g.
negative feedback control loops, and parametrised control systems for
growing organisms, or for use across species.
Erwin Schrodinger in What is life? (1944) argued that biological
reproduction made use of mathematical properties of discrete sequences of
stable chemical structures made possible by quantum mechanisms.
And before that physical and chemical processes of many kinds conformed to
mathematical constraints, e.g. a liquid flowing on a surface will tend to
minimise its gravitational potential energy.
Later evolutionary processes produced mechanisms making more and more
sophisticated uses of mathematics, including brains of many animals, e.g.
squirrels and nestbuilding birds.
Only later did humans not only use mathematical features of the
environment: they also began to think about what they were doing: another
product of biological evolution. (Human toddlers seem to discover and use
topological theorems, unwittingly.)
This talk will introduce some questions about the capabilities of the
universe that made all this possible, providing a different kind of
“foundation” for mathematics: a foundation for mathematical machinery.
This sort of foundation is different from a part of mathematics that
generates the rest.
Such foundational machinery must be a kind of “construction kit” with the
ability to grow an increasingly complex and varied collection of derived
“construction kits” mainly provided by biological evolution, repeatedly
using properties of the fundamental construction kit provided by physics,
to build new more powerful construction kits.
So far nobody has produced a computerbased system capable of making all
the discoveries made by ancient mathematicians. Is that because we are not
clever enough, or could some of the evolved construction kits have features
that cannot be replicated, or accurately simulated, in digital computers—
including features used by animal brains? Finding an answer may require a
multipronged research strategy. I don’t have an answer, yet. But I’ll
suggest a research strategy, within the Turinginspired MetaMorphogenesis
project.
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/metamorphogenesis.html
This talk is part of the Theoretical computer science seminar series.
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