Ernst Dencher
TOK essay outline – F
23/11/08
The strengths and weaknesses of reason as a way of knowing
-
Math – more objective
Bertrand Russell – “Mathematical truth is pure logic and because of this
Mathematics has no subject matter”
based on axioms
gives the truth
not dependent on inductive logic
Deductive
o a priori
 strength:
 weakness:
 only based on our intuition
o which is the basis of all axioms
 no numbers exist until we create them  we create
math(Kant)
o mathematics
 strength:
 when language of math is used purely for mathematicians
as a tool of communication, it is very precise, and
unbewitched
 weakness:
 when natural language is used to communicate
mathematical ideas, it becomes bewitched
o objectivity
 strength:
 free from observation and induction
o Axioms
 Strength:
i. Based solely on what we know
ii. Accepted by all mathematicians
 Weakness:
i. Evidence for axioms may be weak
o Rules of Inference - a priori, rules of inference imply
 Strength:
 Creates new mathematical statements
 Control creation of mathematical knowledge
 reasoning restricted to rules of inference and the original
axioms, therefore allow themselves to build new theorems
 weakness:

it does not give absolute truth, it is merely a conceptual
technique for making explicit what is implicitly contained
in a set of premises – Russell’s Paradox
 fictitious mathematical entities (i.e. complex numbers) does
not excite our imaginations as opposed to other fictitious
elements
o Theorems - stated after applying inference and axioms
 Strength:
 Statements are mathematical knowledge
 Theorems apply to everything, always correct
 Weakness:
 Evidence (source) for the axioms seem vague
Science – more subjective
 The scientific method (naïve inductivism)
(observation  information  generalization  theory  explanation & prediction)
[inductive + deductive]
a. Strength:
i. Can be used to predict similar future occurrences
ii. can give rational explanations for occurrences
iii. best possible form of induction
iv. always improving, finding new solutions
v. (pragmatic) it works, why fix it?
b. weakness:
i. only provides us with information that they observe
(choose to observe)
ii. observations limited to sensual perception of scientists
iii. observations are biased, therefore unreliable/inaccurate/
deceptive
iv. what we really sense is often exactly what we expect to
sense
v. explanation & evaluation is deductive logic so, per
definition, must be correct – however, preceding steps are
based on inductive knowledge (we know to be inaccurate)
vi. induction is ‘probable’, but never ‘certain’
 Falsificationism (Karl Popper)
(hypothesis  state conditions that can disprove hypothesis  test hypothesis
 observation  if falsified: reject/redo or if not falsified: accept provisionally
and wait for future tests)
a. Strength:
i. the certainty that there is no certainty
ii. can distinguish scientific from non-scientific hypotheses
(because non-scientific cannot be falsified)
iii. provides incentive for scientific growth
b. weakness:
i. it is hard to provide new scientific knowledge if everything
is subject to falsification
“Scientists are people with a passion to explain” – Albert Einstein
“Science is an enterprise in which explanations are sought for why phenomena
happen in the way they do. In seeking explanations, scientists use previous
knowledge, new observations, imaginative analogies, and carefully designed
experiments.” - ?