This will involve the rational faculties being in control of the appetitive part of the soul. Pleasure and happiness. According to Aristotle, pleasure is not the aim of every human action, because not every pleasure is good. Remember, the highest good is intrinsically good. Pleasure is found in various forms of activity, and a proper pleasure or pain may belong to any activity.
The pleasure which is found in some forms of activity may be good, and the pleasure which is found in other forms of activity may be bad. Pain may similarly be good or bad. So, pleasure is not the highest good -- it is not the same as happiness.
There are moral virtues and intellectual virtues; we will concerned with the moral virtues. Intellectual virtue comes from teaching, but moral virtue comes from habit.
This means that the two are acquired differently; intellectual virtue can be acquired by reading a book; moral virtue can be acquired only through practice.
An argument:. Nothing can form a habit that is contrary to its nature. Virtues can be formed by habit. Vices can be formed by habit. Therefore, man is neither virtuous nor vicious by nature.
A certain type of situation elicits certain responses in us actions and passions. Depending on how we respond, we will form a habit and become either virtuous of vicious. Thus, vices are acquired by bad habits, just as virtues are acquired by good habits. We become virtuous by acting virtuously. We become vicious by acting viciously.
Virtues are destroyed by defect and excess. They are created by avoiding defect and excess. So too is it, then, in the case of temperance and courage and the other virtues. For the man who flies from and fears everything and does not stand his ground against anything becomes a coward, and the man who fears nothing at all but goes to meet every danger becomes rash; and similarly the man who indulges in every pleasure and abstains from none becomes self-indulgent, while the man who shuns every pleasure, as boors do, becomes in a way insensible; temperance and courage, then, are destroyed by excess and defect, and preserved by the mean.
For moral excellence is concerned with pleasures and pains; it is on account of the pleasure that we do bad things, and on account of the pain that we abstain from noble ones. If you are not virtuous, then virtue is painful and vice is pleasant. This is why it is difficult to become virtuous. On the other hand, if you are virtuous, then virtue is pleasant and vice is painful. So, the more virtuous you are, the easier it is to remain virtuous and become more virtuous.
Note: This article is part 2 of a six-part series on humility written by university faculty. When students first encounter Socrates, it is not uncommon for them to imagine him as a paragon of humility, the philosopher who claimed to know only that he did not know anything. Maybe there was a bit of karma there.
For the most part, though, humble philosophers are not remembered. Still, Socrates is held up as a paradigm for philosophers, challenging claims of knowledge by others and being willing to be challenged in his beliefs. As Plato depicts him, Socrates seldom if ever backed away from a good challenge. As with many things, philosophers are likely to over-process concepts like humility. On the other hand, he who thinks himself worthy of great things, being unworthy of them, is vain; though not every one who thinks himself worthy of more than he really is worthy of in vain.
The man who thinks himself worthy of worthy of less than he is really worthy of is unduly humble, whether his deserts be great or moderate, or his deserts be small but his claims yet smaller. And the man whose deserts are great would seem most unduly humble; for what would he have done if they had been less? The proud man, then, is an extreme in respect of the greatness of his claims, but a mean in respect of the rightness of them; for he claims what is accordance with his merits, while the others go to excess or fall short.
If, then, he deserves and claims great things, and above all the great things, he will be concerned with one thing in particular. Desert is relative to external goods; and the greatest of these, we should say, is that which we render to the gods, and which people of position most aim at, and which is the prize appointed for the noblest deeds; and this is honor; that is surely the greatest of external goods.
Honors and dishonors, therefore, are the objects with respect to which the proud man is as he should be. And even apart from argument it is with honor that proud men appear to be concerned; for it is honor that they chiefly claim, but in accordance with their deserts.
The unduly humble man falls short both in comparison with his own merits and in comparison with the proud man's claims. The vain man goes to excess in comparison with his own merits, but does not exceed the proud man's claims.
Such, then, is the proud man; the man who falls short of him is unduly humble, and the man who goes beyond him is vain. Now even these are not thought to be bad for they are not malicious , but only mistaken. For the unduly humble man, being worthy of good things, robs himself of what he deserves, and to have something bad about him from the fact that he does not think himself worthy of good things, and seems also not to know himself; else he would have desired the things he was worthy of, since these were good.
Yet such people are not thought to be fools, but rather unduly retiring. Such a reputation, however, seems actually to make them worse; for each class of people aims at what corresponds to its worth, and these people stand back even from noble actions and undertakings, deeming themselves unworthy, and from external goods no less. Part of this debate is about pluralism. Roughly, the monists hold that there is a single structure containing all sets, the so-called true universe of sets. The pluralists argue, again roughly, that there is no one structure that should or even can be granted this special status.
Maddy , , has given a career-length argument that set-theoretic monism is the orthodoxy in set theory in the sense that monism can be assumed, pluralism needs to be argued for. Footnote 24 As she convincingly argues, set-theoretic practice was also intended to serve as a foundation for all of mathematics.
Criticism of Maddy has focussed on her claim that set theory today should still be committed to monism Ternullo , Footnote 25 Rittberg , Antos et al. Her argument that set-theoretic monism is the orthodox position in the current set-theoretic pluralism debate is not in doubt. The set-theoretic pluralism debate arose, in part, out of the realisation that some set-theoretic propositions are undecidable from the currently accepted axiom system Zermelo-Fraenkel with Choice, ZFC for short.
A proposition is called undecidable from some axiom system if that axiom system neither proves nor disproves the proposition. Notice that undecidability is a formal notion: set theorists can prove that CH is undecidable from ZFC. Footnote 26 To decide CH thus requires rational argument for such a stronger axiom system. Footnote 27 An undecidable proposition is called absolutely undecidable if there can be no such rational argument.
Because the currently accepted set-theoretic formalisms cannot express absolute undecidability and hence cannot prove that there is no such rational argument, the absolute undecidability of a proposition cannot be proven.
To say that a proposition is absolutely undecidable thus expresses the personal belief that the proposition is beyond our epistemic grasp: our formal and informal epistemic capacities are insufficient to assess the truth-value of the proposition. A central epistemic question of the set-theoretic pluralism debate is about how far our epistemic grasp on set-theoretic propositions reaches: we can prove that there are undecidable statements, but are there absolutely undecidable statements?
The arguments that debating set theorists give for their views on the matter tend to rely on heavy mathematical machinery infused with philosophical considerations which are often reliant on metamathematical and philosophical views.
The outside view is predominant in the current set-theoretic pluralism debate. Here are three examples. One can arguably claim that if this [ Woodin , p. Such a dream solution template, I argue, is impossible because of our extensive experience in the CH and non-CH worlds.
The multiverse view is one of higher-order realism—Platonism about universes— and I defend it as a realist position asserting actual existence of the alternative set-theoretic universes into which our mathematical tools have allowed us to glimpse. Hamkins , p. In the author's opinion, the key methodological maxim that epistemology can contribute to the search for a stronger foundation for mathematics is: maximize interpretative power. Steel , p.
He develops a multiverse logic, which he motivates thus:. We solve this problem by thinking of the domain of set theory as a multiverse of parallel universes, and letting variables of set theory range—intuitively—over each parallel universe simultaneously, as if the multiverse consisted of a Cartesian product [ Footnote 29 ] of all of its parallel universes. The axioms of the multiverse are just the usual ZFC axioms and everything that we can say about the multiverse is in harmony with the possibility that there is just one universe [until a stronger logic is introduced].
But at the same time the possibility of absolutely undecidable propositions keeps alive the possibility that, in fact, there are several universes. He introduces new logical symbols which can express absolute undecidability.
Notice that this allows us to express statements formally, it does not force any convictions about the truth of these statements upon us. Today, set theorists debate whether there are limits to this epistemic grasp, whether there are absolutely undecidable statements. Submission to the currently standard formalisms of set-theoretic reasoning cannot answer this question because absolute undecidability is not traceable by the currently standard formalism.
Instead he adapts them to the problem at hand. In this subsection I argue that the accounts of intellectual humility by Whitcomb et al. Recall that Kidd presented intellectual humility as a two-component virtue.
The intellectually humble are a disposed to take seriously intellectual pressures on their deeply held beliefs and are b willing to revise these beliefs should the pressures demand it. But mere submission to the existing formalisms of set theory cannot handle the issue of absolute undecidability. He does not seek to provide answers to the issue of absolute undecidability.
Thus, he recognises his area of expertise here but is simultaneously willing to adapt it to be able to employ it to the issue at hand. He provides a formal playing field that is free of assumptions about whether there are any absolutely undecidable statements and which they are should they exist.
He does not aim to shape our thoughts on this matter; he allows us intellectual freedom. As they have argued, this is a sign of intellectual humility cf. He did not write a paper and intellectual humility manifested by chance. Rather, he is not disposed to dominate his peers by pushing his views on the pluralism debate upon them. There is a sense in which intellectual humility is a driving force for the writing of the paper.
In this paper I explored how intellectual humility may fail to manifest in mathematical practices. I employed virtue-epistemological accounts of this virtue in three case studies of mathematical activity.
This showed that recent accounts of intellectual humility are successful at tracking some aspects of intellectual humility in mathematical practices but require adjustments in others.
In this section I draw two conclusions from my analysis. Virtue epistemology thus has something to offer to virtue theory of mathematics, namely detailed accounts of the intellectual virtues. On the other hand, what virtue epistemology has to offer is not always successful at tracking the complexities of lived mathematical practices. This was particularly visible in Sect. My suggestion is therefore that virtue theory of mathematics stands to benefit from the intellectual resources provided by virtue epistemology, but these resources may require adjustment to successfully track the virtues as they manifest in mathematics.
This paper suggests that virtue theorists of mathematics ought to appropriate the intellectual resources provided by virtue epistemologists. My second claim in this section is that virtue epistemology is enriched through analyses of the virtues in the context of real life practices. I argue that such analyses can inform us about the nature of the virtues. Additionally, it raises questions about the domain-specificity of the intellectual virtues. Real-life cases of failed manifestations of virtues have characteristically more depth than the pointed thought experiments traditionally relied on in virtue epistemology.
For example, my discussion of the abc-conjecture raised the question for Whitcomb et al. Similarly, the abc-case shows that there can be a remarkable symmetry in intellectual humility: Mochizuki demands from his readers to manifest more humility by getting a better grip on IUT theory, whereas some of his readers demand Mochizuki to better own the limitations of his explanation of the results. That conflicting parties can accuse each other of not being sufficiently humble in their epistemic endeavours is a facet of intellectual humility which has remained unstudied by contemporary virtue epistemology.
Arguably, these insights could have also been obtained from sufficiently smartly constructed thought experiments. The point is, however, that they did not. Thought experiments of the kinds employed in contemporary virtue epistemology are helpful at highlighting issues, but they are bound to oversimplify and thereby overlook facets of the virtue under study that reveal themselves naturally when considering real-life cases.
Studying real-life cases of manifestations of the intellectual virtues can furthermore inform debates in virtue epistemology. Recall that Whitcomb et al.
This theme was taken up again in Sect. In their contribution to this Topical Collection, Tanswell and Kidd question this suggestion. They offer a tripartite distinction: a generic virtues, which are pertinent to all types of enquiry in a domain neutral way; b specific epistemic virtues, which are generic virtues that take domain-specific forms; c local virtues, which are pertinent to a certain subject.
In this paper I have argued that intellectual humility manifests differently in mathematical practices than it does in the conceptual analysis provided by the virtue epistemologists.
However, to claim that intellectual humility manifests differently in mathematical practices than elsewhere implies that mathematical practices are different from other epistemic practices. To argue for such a claim would require a comparative study, which is well beyond the scope of this paper. Suffice to say here that my engagement with mathematical practices focussed on disputes about intellectual ownership, correctness of results, and adaptations of deeply held beliefs, all of which feature in epistemic practices other than mathematics as well.
Whether or not the disputes and adaptations discussed in this paper were informed by any specifically mathematical features that such disputes would lack in other epistemic practices would require further study. This paper thus remains inconclusive on the question whether there are specifically mathematical intellectual virtues.
Virtue epistemology offers valuable theoretical reflections and the philosophy of mathematical practices can provide access to a host of real-life case studies. The development of a virtue theory of mathematical practices thus promises to be beneficial to both intellectual projects.
Noteworthy exceptions can be found in Kidd et al. For a virtue-theoretic analysis of mathematical rigour, see Tanswell This aligns with Whitcomb et al.
Due to lack of space I will not engage with this conception of the virtue. This aligns with Roberts and Wood , p. A noteworthy exception is Kidd In a lecture delivered to the Mathematical Society of Copenhagen. Quotation taken from Goldfeld , p. Published in Goldfeld , p. In his letter to Goldfeld from 6.
Letter printed in Baas and Skau , p. Hardy misjudged this. The impact of the elementary proof of PNT on number theory turned out to be minor. Interestingly, Prudentius already connected humility, pride, and deceit in his Psychomachia ; see Sect. Witnessed by, for example, his proof of a conjecture by Groethendieck on anabelian geometry and his invited talk at the International Congress of Mathematicians in Mochizuki also presents proofs of the strong Szpiro conjecture and the hyperbolic Vojta conjecture.
He has also not openly contributed to the blog discussion between Scholze and Dupuy mentioned above. Note that it is questionable whether monism is the most commonly held position amongst set theorists today. See Rittberg for more on monism as the orthodoxy of contemporary set theory. See Rittberg for an analysis of the interplay between philosophy and set-theoretic activity.
This freedom is nonetheless limited. For an insightful study of rigour as an intellectual virtue of mathematics, see Tanswell Aberdein, A. Virtue Theory of Mathematical Practices. Topical Collection of Synthese. Antos, C. Multiverse conceptions in set theory. Synthese, 8 , — Article Google Scholar. Baas, N. The lord of the numbers, AtleSelberg. On his life and mathematics. Bulletin of the American Mathematical Society, 45 4 , — The virtue of modesty. American Philosophical Quarterly, 30 3 , — Google Scholar.
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