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Seeking certainty in science:
A unity in the experience of the
greatest scientists and philosophers
A not-too-technical view of the roles of
deduction and proof in the history of the physical
sciences, maths & philosophy
Tuesday 23 September 2008
Willi O’Connor
UCD School of Electrical, Electronic and Mechanical Engineering
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Maths 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
Schrödinger’s Equation
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Fascination not new
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Pythagoreans
Plato
Galileo
Descartes
Kant
Positivists
Einstein
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
Mathematics
• Euclid’s Elements, c. 300 BC
• Axioms & theorems
• Deductive method the model in
Maths
…but also in Physics
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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 Philosophy
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Descartes
Kant
Spinoza
Hegel
“The scientific method is the best
and only path to truth”
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 …
• Axioms for numbers in 1899, Peano
• Principia Mathematica, 1910, Russell
& Whitehead: formalisation of proof
• 1931 Gödel’s incompleteness
theorem
Gödel'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 Gödel
• 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?
• Deductive path to certainty not
on offer, not even in Maths
• Still less in Science
• Or Philosophy
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?
• 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?
• 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?
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Sceptical?
Humble?
Indignant?
Arrogant?
Dogmatic?
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”
Aristotle
Nichomachean Ethics
Man as searcher for truth
not certainty
To establish a court of reason, you need a certain, rockhard 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 truth
not certainty
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!
• 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?”
• 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?”
• Anyway, you don’t demand it in life, even
for the most important questions
Finally
• Science is messy, untidy, just like life
• Science defies neat characterisation
• Each “science” has its own methods,
norms, culture, checks & balances
• Scientists themselves are …
Reductionism
• 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.
m1m2
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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”