Justin Clarke-Doane - Morality & Mathematics
Notes to chapter 2: Self-evidence, Proof and Disagreement
Chapter 2 examines the justification of a priori truths in both morality and mathematics, arguing that mathematics is not inherently more self-evident, proof-based, or immune to disagreement than ethics, contrary to common belief.
A widespread assumption is that mathematics relies on a distinct methodology to arrive at (a priori) truths, using axioms and theorems deduced from them. However, this methodology is theoretically transferable to other disciplines. Furthermore, not all axioms are equally 'objective' in Clarke-Doane’s sense of providing unique answers to questions. For instance, the parallel postulate or certain axioms in Group Theory lack this objectivity, while axioms in Arithmetic and Set Theory arguably do. Given the foundational role of Set Theory (and the potential reducibility of all mathematical domains to it), it becomes crucial to focus on it for this analysis.
The consistency of Set Theory's axioms would imply the consistency of other mathematical axioms. However, consistency does not equate to truth; axioms could be consistent yet false. This raises the question: on what grounds can we assert the truth of Set Theory's axioms?
A classical view (e.g., Euclid’s) holds that axioms should be self-evident. Yet, axioms in Set Theory, such as Infinity, Choice, and Replacement (or V=L), are not self-evident, and professional mathematicians often disagree about them.
Inductive approaches, such as Russell's or Rawls’s Reflective Equilibrium, suggest we arrive at axioms based on their coherence with broader systems. This shifts the focus from axioms to the plausibility of propositions derived from them. However, this approach can apply equally to ethical propositions, such as "the death penalty is wrong."
A proposed criterion for justifying mathematical propositions (or axioms) is the relative lack of disagreement about them, especially among experts, compared to moral propositions. However, significant skepticism exists even among experts, not just about Set Theory axioms but across mathematics.
To argue that disagreements in mathematics are fewer than in morality, one might cite widespread agreement on basic propositions (e.g., 2+2=4). Yet, this consensus may reflect pragmatic shorthand or conventional acceptance rather than deep epistemological agreement. Similarly, there are universally appealing moral propositions, such as "don’t burn babies for fun."
The context in which mathematical axioms are taught also complicates their justification. Axioms are often presented as given, fostering consensus that reflects recognition of the "rules of the game" rather than intuitive assent to their truth.
Clarke-Doane also addresses and refutes other arguments for the supposed a priori nature of mathematical truths over moral truths, such as the idea that philosophy distorts initial mathematical intuitions or that mathematical beliefs are analytic.
Given the lack of a clear epistemic distinction between mathematics and morality, why does the belief persist? Clarke-Doane identifies three causes:
Confusion between arithmetic and logic.
Equating logical proof with epistemological truth.
Sociological factors: mathematics enjoys high unanimity in practice, while ethics, being more familiar, emotionally charged, and tied to personal and political consequences, exhibits greater disagreement.
Finally, Clarke-Doane notes that epistemological challenges are not unique to mathematics or ethics but extend to other sciences, including physics and metalogic.
