I criticize a recent paper by Thomas Tymoczko in which he attributes fundamental philosophical significance and novelty to the lately-published computer-assisted proof of the four color theorem (4CT). Using reasoning precisely analogous to that employed by Tymoczko, I argue that much of traditional mathematical proof must be seen as resting on what Tymoczko must take as being "empirical" evidence. The new proof of the 4CT, with its use of what Tymoczko calls "empirical" evidence is therefore not so novel as he (...) maintains. Finally, without attempting to give a full account of the notion of empirical mathematical evidence, I sketch a view showing how the use of calculation injects an empirical ingredient into proof. (shrink)

This paper explores the sense in which rational requirements govern our attitudes like belief and intention. I argue that there is a tension between the idea that rational requirements govern attitudes understood as standing states and the attractive idea that we can directly satisfy the requirements by performing reasoning. I identify the tension by (a) illustrating how a dispositional conception of belief can cause trouble for the idea that we can directly revise our attitudes through reasoning by considering John Broome's (...) view, and (b) advancing a general argument that a standing state cannot be directly affected by reasoning. I then propose a solution: by recognizing the proper targets of rational requirements as occurrent, rather than dispositional attitudes, we can preserve the idea that we can directly satisfy rational requirements through reasoning. (shrink)

To know the meaning of a sentence is to know its truth‐conditions. Yet, in what does the knowledge of truth‐conditions consist? In many cases, it consists in knowing the way in which the truth of the sentence is dependent upon the truth of some other sentences in the language; in some cases, paradigmatically those of the statements of observation. This knowledge consists in the ability to give a report of observation. Then one must know a way. Hence, the notion of (...) truth‐conditions no longer occupies the central place, being replaced by verification and falsification. (shrink)

D. M. Armstrong; II—Does Knowledge Entail Belief?, Proceedings of the Aristotelian Society, Volume 70, Issue 1, 1 June 1970, Pages 21–36, https://doi.org/10.109.

The problem I address is: how are cogent inferences possible? In § 1 I distinguish three senses in which we say that one is “compelled” by an inference: automatic, seductive-rhetorical and epistemic compulsion. Cogency is epistemic compulsion: a cogent inference compels us to accept its conclusion, if we accept its premises and we aim at truth. In §§ 2–3 I argue that cogency is intelligible if we consider an inference as a compound linguistic act in which several component acts are (...) related to one another by a commitment that the premises support the conclusion. Non-automatic inferences are primarily public acts in an intersubjective context. But cogency arises in special contexts described in § 4, epistemic contexts, where the participants care for truth, i.e. are intellectually virtuous. An inference is cogent in an epistemic context if it stands up to all the objections raised in the context. In § 5 I consider three different kinds of cogent inferences. In § 6 I argue that in all three cases cogency is fallible and propose a fallibilist variety of inferentialism. In § 7 I distinguish context of utterance and contexts of evaluation. Cogency is relative to epistemic contexts of evaluation. However, validity, i.e. stable cogency, is transcontextual. (shrink)

This paper reconstructs the Peircean interpretation of Kant's doctrine on the syntheticity of mathematics. Peirce correctly locates Kant's distinction in two different sources: Kant's lack of access to polyadic logic and, more interestingly, Kant's insight into the role of ingenious experiments required in theorem-proving. In this second respect, Kant's analytic/synthetic distinction is identical with the distinction Peirce discovered among types of mathematical reasoning. I contrast this Peircean theory with two other prominent views on Kant's syntheticity, i.e. the Russellian and the (...) Beckian views, and show how Peirce's interpretation of Kant solves the dilemma that each of these two views faces. I also show that Hintikka's criterion for Kant's synthetic judgments, i.e. a new individual introduced by the -instantiation rule, does not capture the most important characteristic of Peirce's theorematic reasoning, i.e. the process of choosing a correct individual. (shrink)

The theory of meaning may either attempt to explain what it is to have the concepts expressible in a particular language, or merely to associate concepts with words in that particular language. A theory of meaning suggested by Davidson, according to which knowing the meaning of a sentence consists in the ability to produce a T‐schema for that sentence, is a modest theory. But such a theory will be either a translation manual, or will have to be construed holistically.

In the last few decades there has been a revival of interest in diagrams in mathematics. But the revival, at least at its origin, has been motivated by adherence to the view that the method of mathematics is the axiomatic method, and specifically by the attempt to fit diagrams into the axiomatic method, translating particular diagrams into statements and inference rules of a formal system. This approach does not deal with diagrams qua diagrams, and is incapable of accounting for the (...) role diagrams play as means of discovery and understanding. Alternatively, this paper purports to show that the view that the method of mathematics is the analytic method is capable of dealing with diagrams qua diagrams, and of accounting for such role. (shrink)