Is mathematics just like any other science?
Lakatos on the mechanics of theory revision
At the beginning of the 20th century mathematics was in crisis. More and more paradoxes were discovered, such as Russell’s paradox proving that naive set theory is contradictory. Several competing schools tried to end the foundational crisis of mathematics with their proposals: logicism, formalism, and intuitionism.
But how should we choose a foundation? How do we choose the right mathematical theory?
In his article “A Renaissance of Empiricism in the Recent Philosophy of Mathematics”, the philosopher Imre Lakatos argued that the mechanisms of foundational theory choice in mathematics are very much like those for choosing a theory in the natural sciences.
How do we choose a theory in the sciences? One crucial way of doing this is to analyse whether a given theory is falsified.
Consider, for example, a (very naive) physical theory predicting that every object, when dropped from a height of n meters reaches the ground in n second.
This theory is false. Why? Because it doesn’t accord to our observations of the physical facts: If I climb the roof of my house and drop an object from 10 meters high, then I will observe that it reaches the ground in much less than 10 seconds.
So it seems that the potential falsifiers of physical theories are all the observations we can make about the facts of the physical world by measuring time, distance, mass, et cetera. The potential falsifiers of science are those facts that have the potential to falsify a scientific theory.
If a scientific theory is falsified again and again, then scientists will discard it. It’s not a good theory if it doesn’t match our observations.
Is mathematics just like the other sciences?
Lakatos thinks that mathematicians are much more empirical than they think. He illustrates this with quotes of famous mathematicians. For example, Lakatos quotes Fraenkel asserting that:
the intuitive or logical self-evidence of the principles chosen as axioms [of set theory] naturally plays a certain but not decisive role; some axioms receive their full weight rather from the self-evidence of the consequences which could not be derived without them. (Fraenkel as quoted by Lakatos)
In this quote, Fraenkel argues that not the axioms that are obviously true but rather that we deem the axioms true in virtue of their consequences — if an axiom has good consequences, then we adopt it for use in mathematics.
Lakatos also quotes Gödel, who makes a very similar point when he discusses foundational theories in mathematics. Those are theories like Zermelo-Fraenkel set theory with Choice, ZFC, that mathematicians use as a basis for their arguments. Gödel says that:
the role of the alleged ‘foundations’ is rather comparable to the function discharged, in physical theory, by explanatory hypotheses . . . The so-called logical or set-theoretical ‘foundation’ for number-theory or of any other well established mathematical theory, is explanatory, rather than really foundational, exactly as in physics where the actual function of axioms is to explain the phenomena described by the theorems of this system rather than to provide a genuine ‘foundation’ for such theorems. (Gödel as quoted by Lakatos)
Gödel points out that axioms explain rather than provide a foundation. We have an informal conception of mathematics — for example, when we calculate on paper, or conduct informal proofs in analysis — and the axioms we find should explain these established mathematical theory.
Mostowski is quoted by Lakatos as follows:
[Gödel’s] and other negative results confirm the assertion of materialistic philosophy that mathematics is in the last resort a natural science, that its notions and methods are rooted in experience and that attempts at establishing the foundations of mathematics without taking into account its originating in the natural sciences are bound to fail. (Mostowski as quoted by Lakatos)
This is a very strong statement: Mathematics originates in the natural sciences and mathematical methods are confirmed by experience.
Potential falsifiers in mathematics
If mathematics is anything like the other sciences, then there must be potential falsifiers in mathematics, i.e. statements that have the potential to falsify a mathematical theory. What could those be?
Lakatos argued that there are two kinds of falsifiers for mathematical theories: potential logical falsifiers and potential heuristic falsifiers.
A potential logical falsifier is the derivation of a contradiction in a mathematical theory. Why does this falsify a mathematical theory? Remember Russell’s paradox. He showed in 1901 that naive set theory is inconsistent by providing the definition of a set that leads to contradiction: Take y to be the set of all sets that do not contain themselves. Does y contain itself? If it does, then it doesn’t. And if it doesn’t, then it does.
This observation is a logical falsifier. It shows that naive set theory cannot be right because it is contradictory.
And yes, the discovery of this logical falsifier led mathematicians to abandon naive set theory. Instead, more intricate set theories like Zermelo-Fraenkel set theory ZF were developed.
Lakatos also discusses potential heuristic falsifiers:
But if we insist that a formal theory should be the formalisation of some informal theory, then a formal theory may be said to be ‘refuted’ if one of its theorems is negated by the corresponding theorem of the informal theory. One could call such an informal theorem a heuristic falsifier of the formal theory. (Lakatos)
The idea here is that formal mathematical theories serve as a formal underpinning for the informal mathematics we have been doing all along: what a foundation should do is to make clearer why and how our theorems are true, and give us confidence in the theorems we already know.
Let’s conduct a thought experiment. Imagine you are trying to develop a formal mathematical theory to account for arithmetic. That is, you want to find a ‘formal underpinning’ to make explicit how and why certain facts of arithmetic are true. Imagine, moreover, that you have a proposal for such a theory. Call it T. Playing around with your theory for a while, you suddenly notice that it proves that ‘2 + 3 = 4.’
That’s weird and unwanted. Your goal was to find a theory that explains how arithmetic works in practice. But what you came up with is a theory that does not get the basic facts of arithmetic right: T proves that 2 + 3 is 4 but that’s not right because informal arithmetic, just calculating with our fingers, tells us that 2 + 3 is 5.
Lakatos would say that this theory T is heuristically falsified or even arithmetically falsified.
Conclusions
I hope that this gave you a good impression of how Lakatos tries to show that theory revision in mathematics is very similar to theory revision in the sciences: His argument is that mathematics, just like the sciences, has potential falsifiers that could falsify mathematical theories. And if a theory is falsified, then it has to be revised. This happens in mathematics just like in the sciences.
Of course, mathematical falsifiers are somewhat different than the falsifiers of, say, physics. Still, Lakatos’s examples indicate that mathematics is more similar to the ‘normal sciences’ than one might think.
If my article sparked your interest, I can recommend reading Lakatos’s original article in full: A Renaissance of Empiricism in the Recent Philosophy of Mathematics.
What do you think? Is mathematics just like any other science? Or is it special? Feel free to tell me in the comments. And if you’re curious for more stories on logic and mathematics, have a look at my profile.
