Wittgenstein and Mathematical Truth

Hello! Welcome to the final lesson in our module on Contemporary Philosophy of Mathematics.

In our previous lessons, we explored realist views like Platonism and Structuralism, which treat mathematical truth as a matter of discovering facts about an objective, mind-independent reality. Today, we turn to one of the most radical and influential critiques of this entire picture, coming from the later work of Ludwig Wittgenstein.

Your learning outcome for this lesson is to: Analyze how Wittgenstein's rule-following considerations challenge traditional views on mathematical truth.

Instead of asking what mathematical objects are or where they exist, Wittgenstein forces us to ask a more fundamental question: What does it mean to follow a mathematical rule at all? His investigation into this question unravels the very notion of objective mathematical truth that Platonists and others take for granted. We will see how he proposes to replace the idea of "correspondence to reality" with "agreement in a form of life."

1. Mathematics: Invented or Discovered?

Traditional realism, particularly Platonism, views mathematics as a process of discovery. The mathematician is like an explorer charting a pre-existing, abstract landscape. Wittgenstein turns this idea on its head.

To begin, let's watch a short video that introduces Wittgenstein's core thesis: that mathematics is fundamentally a human invention, not a discovery.

As the video explains, Wittgenstein rejects the idea that mathematical statements are true because they correspond to some abstract reality. Instead, he sees mathematics as a human activity, a set of language-games we have created. For him, a statement like "2+2=4" is not a description of a fact in a Platonic heaven; it's a rule that governs how we use symbols.

This anti-realist stance is a constant theme in his later work.

This shift from discovery to invention is the first step. The second, more devastating step in his argument comes from his analysis of what it means to follow the rules we invent.

2. The Rule-Following Paradox

If mathematics is a system of rules, then a mathematical calculation is simply the correct application of those rules. But what guarantees that our application is correct? What does it even mean to follow a rule?

This leads to Wittgenstein's famous rule-following paradox. The following video provides an excellent explanation of this puzzle, using a now-classic mathematical example developed by the philosopher Saul Kripke in his influential interpretation of Wittgenstein.

Let's break down the paradox, which will be familiar territory given your background in formal systems.

Imagine you are asked to compute 68 + 57. You answer 125. But a skeptic challenges you: "How do you know you were meant to add? In the past, you've only ever performed a finite number of additions. All your past actions are consistent with the rule of quaddition (symbolized by '⊕'), which is defined as:
$x\oplus y=\{\begin{array}{ll}x+y& \text{if }x,y<57\\ 5& \text{otherwise}\end{array}$
Since you've never added numbers this large before, how do you know you were following the + rule and not the ⊕ rule? If you were following the ⊕ rule, the correct answer would be 5."

Your first instinct might be to appeal to the definition of addition, perhaps by breaking it down into successive 'add one' operations. But the skeptic can simply apply the same logic: how do you know you are following the 'add one' rule and not the 'quadd one' rule?

The paradox is this:

"No course of action could be determined by a rule, because every course of action can be made out to accord with the rule." (Philosophical Investigations §201)

There seems to be no fact—not in the external world, not in my mind, not in my past actions—that can definitively establish that I mean addition rather than quaddition. The "rails of logic" that Platonism imagines stretching to infinity, pre-determining every correct answer, seem to vanish upon inspection.

3. The Challenge to Traditional Mathematical Truth

The rule-following paradox directly attacks the foundations of traditional views of mathematical truth.

• For the Platonist: The truth of "68 + 57 = 125" comes from a fact about the abstract objects 68, 57, and 125, and the addition function. Wittgenstein's paradox suggests that the very meaning of "addition" is not fixed by any such fact. There is no Platonic "super-fact" that makes it true that the + sign refers to addition and not quaddition.
• For the Mentalist: One might argue the meaning is fixed by a mental state or idea. But the same problem applies. What fact about my mental state determines that it is an "idea of addition" and not an "idea of quaddition"? Any mental representation is just another sign that can be interpreted in multiple ways.

So, where does this leave us? Is mathematics just arbitrary? No. Wittgenstein's move is to dissolve the paradox by showing it arises from a mistaken picture of what meaning and understanding consist in.

A "Skeptical Solution": Community and Forms of Life

Correctness in following a rule doesn't come from correspondence to a Platonic fact or a private mental state. It comes from being a trained member of a community that shares a "form of life".

The key points are:

1. Meaning is Use: The meaning of + isn't a mystical connection to an abstract function. It's the role the symbol plays in our practices: in counting, measuring, building, and so on.
2. Training, not Interpretation: We don't interpret rules; we are trained to react to them in specific ways. When you see 68 + 57, you don't interpret the +; you react in the way you were drilled, just as a trained soldier reacts to a command. You obey the rule "blindly."
3. Agreement in Action: Mathematical truth is grounded in the fact that, after training, we nearly all agree in our actions and judgments. We call 125 the right answer because that is how our community goes on. This isn't an "agreement in opinion" (which would be subjective conventionalism), but a much deeper, pre-rational agreement in form of life. The fact that we can be trained to agree in this way is a brute, contingent fact about us as biological and social beings.

Without this background of regularity in the world and in our reactions, mathematics as a practice could not exist. The "peculiar inexorability" of mathematics comes not from a transcendent realm, but from the fact that these rules have been "hardened" from empirical regularities and now function as the fixed framework by which we judge experience.

Conclusion

Wittgenstein's rule-following considerations mount a profound challenge to traditional philosophies of mathematics. By questioning the very ground of meaning, he seeks to replace a metaphysics of objects and abstract realities with an anthropology of human practices.

Key Takeaways:

• Wittgenstein argues that mathematics is a human invention, not a discovery of pre-existing truths.
• The rule-following paradox shows that no objective fact (Platonic, physical, or mental) determines the correct application of a rule, undermining the idea that mathematical truth is about correspondence to a determinate reality.
• Wittgenstein's "solution" is to ground mathematical correctness in shared practices, training, and "agreement in form of life" within a community.
• On this view, a mathematical proof does not reveal a pre-existing truth. It creates a new rule or connection, expanding our mathematical language-game and giving new meaning to the symbols involved. Mathematical truth is therefore not static and eternal, but something that evolves with our practices.

Preview of the Next Module:
This concludes our deep dive into the philosophy of mathematics. We have journeyed from Platonism's abstract objects to Fictionalism's useful stories, and finally to Wittgenstein's community-grounded rules. In our next module, "Contemporary Analytic Metaphysics and Epistemology," we will broaden our scope but revisit some familiar themes. We will begin by exploring the problem of universals, which asks what properties (like 'redness' or 'primality') are, and whether they exist independently—a question that will surely echo the debates we have just had.