Seeking certainty in science: Seeking certainty in science: A unity in the experience of the A unity in the experience of the

greatest scientists and philosophersgreatest scientists and philosophers

A not-too-technical view of the roles of A not-too-technical view of the roles of deduction and proof in the history of the physical deduction and proof in the history of the physical

sciences, maths & philosophysciences, maths & philosophy

Tuesday 23 September 2008Tuesday 23 September 2008

Willi O’ConnorWilli O’Connor UCD School of Electrical, Electronic and Mechanical EngineeringUCD School of Electrical, Electronic and Mechanical Engineering

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Maths modelling natureMaths modelling nature

“How can it be that mathematics, being after all the product of human thought independent of experience, is so admirably adapted to the objects of reality?”

Einstein, 1920

SchrSchröödinger’s Equationdinger’s Equation

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Fascination not newFascination not new• Pythagoreans

• Plato

• Galileo

• Descartes

• Kant

• Positivists

• Einstein

Hierarchy among the sciences?Hierarchy among the sciences? • Soft to hard, qualitative to quantitative,

approximate to exact

• More “scientific”

• Within disciplines also

• Also computer science

• Within Engineering: maths & philosophy!

“All Science is either Physics or stamp collecting.”

Rutherford, 1871-1937

MathematicsMathematics

• Euclid’s Elements, c. 300 BC

• Axioms & theorems

• Deductive method the model in Maths

……but also in Physicsbut also in Physics• Newton’s 3 laws of motion & gravity• Thermodynamics• Electromagnetism• Quantum Mechanics• Relativity• String theory• Theories of everything

“Hypothetico-deductive” or

“Deductive-nomological” method

… … and in Philosophyand in Philosophy

• Descartes

• Kant

• Spinoza

• Hegel

“The scientific method is the best and only path to truth”

Meanwhile back in Maths …Meanwhile back in Maths …• Euclid 23 centuries

• Consistency? Completeness? Absolute proofs of consistency?

• Hilbert 1842-1943: Hilbert’s program: proof by formal rules for manipulating formulae

“Proof is the idol before whom the pure mathematician tortures himself.” Eddington, Astronomer (1882 - 1944), The Nature of the Physical World

Meanwhile back in Maths …Meanwhile back in Maths …

• Axioms for numbers in 1899, Peano

• Principia Mathematica, 1910, Russell & Whitehead: formalisation of proof

• 1931 Gödel’s incompleteness theorem

Gödel's Incompleteness TheoremGödel's Incompleteness Theorem

• Given any consistent set of arithmetical axioms, there are true mathematical statements that cannot be derived from the set

• There are always some propositions that can't be proven either true or false using the rules and axioms

Implication of GImplication of Göödeldel

• A deductive system in Arithmetic with an absolute proof of consistency now looks impossible.

• There are endless true arithmetical statements that cannot be formally deduced from given axioms by set of rules of inference.

So what?So what?

• Deductive path to certainty not on offer, not even in Maths

• Still less in Science

• Or Philosophy

So what?So what?

• “There’s no point even discussing that because it can’t be proven”

• (Also attitude to philosophy)

Response from an unexpected source …

We shall never have done beginning if we determine to begin with proof. We shall ever be laying foundations, … We shall never get at our first principles. Resolve to believe nothing and you must prove your proofs and analyze your elements, sinking farther and farther, and finding 'in the lowest depth a lower deep', till you come to

the broad bosom of scepticism. Newman, 1870 Grammar of Assent

Instead of devising what cannot be, some sufficient science of reasoning which may compel certitude in concrete conclusions, confess that there is no ultimate test of truth besides the testimony borne to truth by the mind itself, and that this phenomenon, perplexing as we may find it, is a normal and inevitable characteristic of the mental constitution of a being like man on a stage such as the world. His progress is a living growth, not a mechanism: and its instruments are mental acts, not the formulas and contrivances of language.

Newman, 1870 Grammar of Assent, p. 275

As in Maths we're justified by dictate of nature to withhold our assent from a conclusion of which we have not yet a strict logical demonstration, so by a like dictate we are not justified, in the case of concrete reasoning …, in waiting till such logical demonstration is ours, but on the contrary we are bound in conscience to seek truth and to look for certainty by modes of proof which, when reduced to the shape of formal propositions, fail to satisfy the serene requisites of science.

Newman, 1870 Grammar of Assent

Language, logic, formal systems of reasoning, including Science and Maths, however useful in extending and improving the mind's ability to seek and find truth, are ultimately only instruments of the mind, whose only sanction and validity come from other mental acts.

Newman, 1870 Grammar of Assent

So what?So what?• Truth is a broader concept than certainty• There is no algorithm, syllogism, or

formulaic procedure which establishes certainty on any topic. Ultimately we simply "see" that something is so, is true.

• The algorithm, syllogism, or formulaic procedure are themselves things that must be seen, that depend on the mind’s seeing them as true / valid.

Is this a problem?Is this a problem?• Only if we decide it is! (Descartes decided)• Not to have certainty should not undermine

reason. To discover limits to reason is to use reason at its highest level.

• Godel’s theorem is a truth, a discovery, a use of ingenuity escaping / mastering the dull mechanics of logic.

• Heisenberg also a profound achievement of the mind, a contribution to knowledge, to truth.

What attitude to take?What attitude to take?

• Sceptical?

• Humble?

• Indignant?

• Arrogant?

• Dogmatic?

What attitude to take?What attitude to take?

• Humble or arrogant to demand certainty?• What is extraordinary is that our minds can

create maths, and physics, and science• & that the universe is understandable, to

any degree, by the human mind. • Why demand that the universe fit inside

our heads! Why should it? And then why should we have certainty? (mathematically defined, at that)

“The most incomprehensible thing about the universe is that it is comprehensible.”

Einstein

““Man must philosophise”Man must philosophise”

AristotleAristotle

Nichomachean EthicsNichomachean Ethics

Man as searcher for truthMan as searcher for truth notnot certaintycertainty

To establish a court of reason, you need a certain, rock-hard foundation to start from. None exists. Experience of Descartes & successors: on a mental peg you can hang only a mental coat.Being is not founded on the mind: the mind is founded on being.

If reason is doubted, what higher tribunal can you appeal to? (without using reason, and appealing to reason?).

Man as searcher for truthMan as searcher for truth notnot certaintycertainty

Crisis of Certainty vs scepticism: Crisis largely connected with desire for certainty.Temptation to make man measure of everything,

and if certainty not available, all is subjective, relative, arbitrary

Contradiction obvious, but attitude remains

Freedom!Freedom!• The mind reserves the right to form judgments

independently of the tyranny of logic, the straightjacket of a purely mechanistic process.

• “There must be freedom in the theoretical acts of affirmation and negation. When I reason that 2+2=4, this actual judgment is not forced upon me through blind, natural causality” Hermann Weyl, 1932, Mathematician

• Rational thought cannot be entirely determined by formal rules, nor by purely physical factors.

• Haldane “If my thought processes are determined by chemical processes going on in my brain, they are determined by chemistry, not reason or logic.

• If all my thought processes are simply motions of electrons in my brain, I’ve no reason to believe they’re true.

• So I’ve no reason to believe all my thought processes are just motions of electrons in my brain.

““But can you prove it?”But can you prove it?”• The concepts “certainty” and “proof” are

not as simple as they seem. • You can always define “certainty” invoking

reasonable criteria of proof such that nothing will be provable by those criteria.

• Absolute certainty (in some mathematical or logically watertight sense) is not on offer, on any topic, ever. So why are you demanding it here?

““Can you prove it?”Can you prove it?”

• Anyway, you don’t demand it in life, even for the most important questions

FinallyFinally

• Science is messy, untidy, just like life

• Science defies neat characterisation

• Each “science” has its own methods, norms, culture, checks & balances

• Scientists themselves are …

ReductionismReductionism

• Can all Science be reduced to Physics?

• Maths models reality, it is not reality. It models quantifiable, measurable, aspects, of physical reality, that’s all

• Schrodinger’s equation of the electron is not the electron. Maxwell’s equations are not electromagnetism. Einstein’s equations are not Gravity.

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Important questions missed!Important questions missed!

All the things men feel passionate about such as justice, integrity, courage, love, reason ... and all the ultimate questions, like Why life? Why the universe? Why do I exist? What is man? What is good/evil?... are all ignored by Science, systematically.

“It is not just that the world is stranger than we ever imagined: it is even stranger than we can imagine.”

“Wow”