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

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• Aaron Sloman (University of Birmingham)
• Friday 11 November 2016, 11:00-12:00
• Computer Science, The Sloman Lounge (UG).

If you have a question about this talk, please contact Paul Taylor.

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/?go=overview http://theorymine.co.uk/?go=about [The main author, Alan Bundy, will visit us next term.]

(In a later talk I’ll discuss some apparent limitations in current AI theorem proving technology.)

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. pure 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 the future sequence of human brains?)

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.

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-insired Meta-Morphogenesis project. http://www.cs.bham.ac.uk/research/projects/cogaff/misc/meta-morphogenesis.html

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

This talk is included in these lists:

• Computer Science Departmental Series
• Computer Science Distinguished Seminars
• Computer Science, The Sloman Lounge (UG)
• Theoretical computer science seminar
• computer sience

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