Physical/evolutionary foundations for mathematics vs logico/semantic foundations for mathematics

Aaron Sloman -- School of Computer Science

Notes for an informal seminar in the School of Computer Science, University of Birmingham Friday 11th September 2016

See also:

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 non-human, 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 non-mathematicians realise.

The abilities are "layered", like many products of evolution.

AI (including robotics) is currently seriously limited by lack of a theory explaining wide-spread, 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 nest-building 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 pre-verbal human toddlers acquiring and using theories about 3-D 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 meta-cognitive 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 "non-standard" 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

  • acquiring using and storing energy
  • acquiring using and storing materials for construction and repair
  • reproductive processes

    See Tibor Ganti The Principles of Life

The physical universe also produces lots of non-living 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 information-processing 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:

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

(Nothing in AI so far can replicate the discoveries of Euclid, Archimedes, etc. or a pre-verbal human toddler exploring 3-D topology.)

( I have several online papers on limitations of current AI technology, e.g.

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:

(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 counter-evidence, 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 nest-building 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 computer-based 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 multi-pronged research strategy. I don’t have an answer, yet. But I’ll suggest a research strategy, within the Turing-inspired Meta-Morphogenesis project.

This talk is part of the Theoretical computer science seminar series.

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