1

Remark: my question deals more particularly with the axioms of set theory, arithmetic, probability theory, etc. I think the status of the axioms in geometry is clearer.

The French fictitious mathematician Bourbaki writes somewhere ( " The Architecture of Mathematics " in Jean-François Le Lionnais, Great Currents Of Mathematical Thought ) that the approach of the mathematician is comparable to the way experimental science proceeds.

The physicist makes observations and looks for the best explanation: he adopts as his theory the hypothesis ( or set of hypotheses) from which these observations can be deduced at minimuml cost.

If the comparison holds, the " observations" would, in mathematics, be some pieces of mathematical knowledge the mathematician wants to " secure" or justify , and the axioms would be the best available explanation. For example, the mathematician first wants addition to be commutative, multiplication to distribute over addition, etc., and after that, he seeks hypothesis or axiom(s) from which this desired results could follow.

My question is : is this view of axiomatizing in mathematics correct? and could this conception of axiomatizing helpfull to correct the feeling of gratuitousness or arbitrariness of axioms?

3

To state it short, no. The two are not comparable.

Now for the long winded version: the question is simultaneously relevant to two branches of philosophy: philosophy of science and philosophy of mathematics.

A short primer on philosophy of mathematics:

  1. Mathematical realism is the view that mathematical entities actually exist and are "real" entities, either abstract (platonism), or concrete (mathematicism).
  2. Anti-realism is the view that there are no mind-independent mathematical entities.

A short primer on philosophy of science:

  1. Realism within science is the view that science provides us with the knowledge of THE world. That is, THE mind-independent externality.

  2. Anti-realism is the view-point in which science does not, in fact, tell us anything about THE world, but it merely gives us the means to organize our sense-data.

Bearing that in mind, we can necessarily agree on one thing: Mathematical facts, or to be slightly technical, theorem, regardless of what philosophical school you belong to, are objective. That is, Mathematical statements, if proven true, are objective and unchanging. Its reason, however, of being objective are open to debate. Realists will argue it is true because when we engage in mathematics, we engage in discovery of real abstract entities which do not change. Anti-realists, however, would argue the statements, if they are a nominalist, are true solely because that is how we have defined our system of mathematics.

Where does a scientific hypothesis fall then?

  1. If you are Kuhnian (referring to thomas Kuhn), then for you a scientific hypothesis is really an axiom. That is, of course, only to the point you start experiencing unresolvable anomalies.
  2. If you are a realist, then scientific hypothesis describe the world, but they are never certain (refer to problem of Induction by Hume)(Goodman's new problem of induction argues confirmation can never be formalized since it is necessarily semantic). Which, obviously implies, they are incomparable to axioms, because mathematical axioms are never uncertain; that is, they are never brought into question.
  3. If you are an anti-realist, then scientific hypotheses are mere tools for you which can be discarded at any point in time (refer too instrumentalism). This, too, implies mathematical axioms and scientific hypotheses are incomparable.

Summary

Scientific hypotheses and mathematical axioms are vastly different. One is necessarily certain, while the other necessarily uncertain. They are comparable to one another only because we deduce further information from both. However, this similitude breaks in close scrutiny as well.

I hope that answered your question, feel free to ask for any clarification.

  • Axioms, in simplest terms, are statements which can be easily and widely accepted but can't be proven. They then stand as a starting point for any given hypothesis which should be held to some standard of proof. Is it just me who finds your juxtaposition of 'real' and 'abstract' very humorous? CS – Charles M Saunders Mar 28 at 14:51
  • @user37981 Right. – Bertrand Wittgenstein's Ghost Mar 28 at 15:26
  • @user37981 Not sure why you would find it humorous, any particular reason? – Bertrand Wittgenstein's Ghost Mar 28 at 17:04
  • Sorry, it's just that Spinoza abhorred abstraction, so in the parlance I'm familiar with, real and abstract are oxymoronic. @BertrandWittgenstein'sGhost – Charles M Saunders Mar 29 at 16:01
  • @user37981 No problem. Continental Philosophy is not my cup of tea. – Bertrand Wittgenstein's Ghost Mar 29 at 20:08

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