Hello everyone, and welcome to this talk about global balance on lattice saving and information set decoding. This is joint work with Elena Kishanova, and this talk is for Crypto 2021. Okay, so as we all know, letters-based and code-based cryptography are two of the prime candidates for designing post-quantum cryptography, which is hopefully secure against quantum attacks. So, for instance, in the NIST standardization process, which is going on right now, several of the finalists are based on lettuces and codes. And of course, the security of these letters-based and code-based cryptographic schemes relies on the hardness of the underlying letters problems and decoding problems. So it is important that we study these problems well, and that we study the algorithms for these problems well, so that we know the security of these cryptographic schemes and how to browse. And note that as we are usually interested in long-term security, we would ideally like to have conservative bounds on the attack cost. Like, we would like to account for possible improvements in the hardware that people just have faster algorithms in the future because their computers get faster. But also, maybe someone will come up with better analytics techniques, better algorithms to solve these problems. Ideally, we would also account for that with our parameter selection now, so that in 10 years, like, security does not have to be lower than we hoped for, that we aimed for. So the only way to really try to solve this problem is to look at the state-of-the-art script analysis, like the best algorithms for solving these hard problems and see if they can be improved or not. And so in the context of lettuces, letter sieving combined with a basis reduction, that is, the basis reduction is currently, like, the state-of-the-art. And in code-based cryptography, like, information decoding algorithms are some of the most important tools. And both of these approaches kind of follow a similar approach at some point, where they sample a large database of long vectors, where long is long under some metric, which I will get to later. So we sample a lot of these vectors. Then we combine nearby vectors to obtain shorter vectors. So we have this whole list of vectors, and now we try to find a search for pairs of vectors that we can combine, say, by adding them or subtracting them or something, something simple at least, so that we get a nicer, shorter vector. And we only want to combine vectors, like, we can only combine vectors which are nearby in space. So again, nearby is under the same metric of long and short here. And then we just repeat this of combining vectors until we find many short vectors. So we start with the large database of long vectors, and then we iteratively like to find shorter and shorter vectors. Until at the end, we have a bunch of short vectors. And then, for instance, we can solve the shortest vector problem on letters by just doing this several times. And then at the end, we find issues in our list. So a key sub-routine in this approach in both settings is the closest pairs problem, where you have a list of vectors in your space, say, in this database, in this case. And we want to find almost all pairs of vectors that can be combined to find shorter vectors. So we want to find all pairs of nearby vectors in our list. And then this sub-routine dominates the overall algorithm complexities, like if we can improve the closest pair algorithm, then we can improve the overall complexity of letters saving and for information set decoding algorithms. And note that naively, you can just find all pairs of vectors by going through all pairs of vectors. So there will be like quadratic time in the side of the database. But there are these smarter approaches, which also work in high dimensions. So for instance, these hash-based approaches, they have a sub-primary complexity, like depending on how nearby you want to find them. But usually they are like values of quadratic in the side of the list. And right now there are no matching lower bounds, neither for letters saving or for information set decoding setting. So there might be improvements still possible, which is kind of problematic if we want to choose parameters now, if someone comes up with further improvements in 10 years, then there will be a problem. Are improvements still possible, or can we rule them out now? So the main contribution of this paper is to try to study lower bounds for the corresponding nearest neighbor problems that you want to solve when you solve this closest pair problem. Our lower bounds are conditional on the hash-based model, which you will get soon. They don't necessarily apply to all approaches for solving closest pairs problem, but at least within this model, which is quite broad, like it covers almost all approaches, at least like for saving to date. So if you are in this model, then this lower bound applies. And for the letter saving setting, this lower bound is actually tight. Like it matches the current known best techniques for solving this nearest neighbor problem. So for instance, this gives you conditional lower bounds that result from 2016 with a 0.292 exponent for SCP is actually optimal in this model. And also for information and decoding, we get lower bounds, but these are unfortunately not asymptotically tight. They are almost tight, but not quite tight. So this could, for instance, mean that the best known, best currently known nearest neighbor technique for my and Oso from 2015 is actually suboptimal. What could be that just our lower bound is not good? So it could be that there's a better lower bound, which actually matches this result of 2015. So it's not really clear which of the two is the case, but we do want to stress actually the gap between the lower bound and the upper bound is quite small. So there might be some small improvements in the future, but not really that much. Low bound is quite close to the upper bound. And so, yeah, what are the implications of these results? So mainly this gives you a better understanding of the hardness of the underlying problems, like the lattice problems and the decoding problems. And so for the cryptanalysis, if you're working on cryptanalysis, especially in the lattices part, then probably the takeaway message here is that you should search for improvements elsewhere, like not in the nearest neighbor technique anymore, like this kind of, this lower bound kind of solves the problem in that the lower bound matches the upper bound. So maybe you can find improvements in other parts of lattice saving or in other lattice algorithms in general. This does motivate further work on the best lattice shift, like this has already been done recently, like studying like this result from 2016, studying this algorithm and looking at like the specification overhead or how you implement this efficiently, like on GPUs for instance, and seeing what you can get for the performance here, because like now we know that this result from 2016 is actually optimal, at least within this model. So it makes sense to really zoom in on this particular approach. And if you are working on, say, designing cryptography and you want to choose your parameters, then yeah, what is the result? What our contributions tell you is that you have some kind of conditional security guarantee, like if an attacker is trying to attack your scheme by, say, solving the underlying lattice problem and they are solving this lattice problem by running, say, lattice saving with some hash-based nearest neighbor technique, then they're not going to better than what is already known from 2016. So the attacker would really have to improve some other part of the algorithm to really break your scheme and decrease your security. Okay, so I will not cover all the results in this talk, in the remainder of this talk. So I will just briefly try to sketch the lattice saving contribution because that's arguably the nicest result that we have. So just to sketch bit formally, like the closest pairs problem, so we are given some bounded metric space, we have the m with a distance d and we have a distance metric d and we have a target distance r, like which qualifies when vectors are nearby. And we have a list l which is a subset of m with elements drawn uniformly at random from m. So we are not considering worst case settings, we are considering average case settings where l is really random from m. And the problem asked to find almost all pairs x, y and l such that the distance is at most r. And almost all here means that present you want to find 90% of them or 95% or like you can miss a constant fraction but you cannot miss too many of them. But you also don't need to find all of them, like if you find 90% of them, that's already quite good. And so one of the main approaches for solving this problem uses locality-sensitive hash functions which are functions h here, which satisfy this property that if you have two vectors in your metric space, say x, y and m, and their distance is small, then the probability that the hash values are the same, so that the hash vector is equal to the hash of y is much larger than if x and y were just drawn uniformly at random from m without this guarantee on the distance between x and y. These hash functions are locality-sensitive in the sense that if two vectors are nearby in space, then the hash values will be more similar than if they are random in space. So if you have such hash functions, like if you have a big gap between the left-hand side and the right-hand side, then you can design an algorithm around this by building and populating hash tables using these hash functions. So for a vector x, you store it in a hash bucket labeled the hash effects, and for a vector y, you store it in the hash bucket labeled the hash of y. You do this for all vectors, and then within each hash bucket, you combine pairs of vectors. You try to combine them to make progress. You hope that these are closed pairs, and because this hash function has this property, ideally this would then give you a faster algorithm than going through all pairs of vectors. If you want to instantiate this completely, like for letter-saving, the space is the unisphere, like this abstraction, this heuristics, based on that you have points in the unisphere, and then the distance is like a Euclidean metric, but the distance is all the same on the unisphere, so you can also consider the angular distance by having a large load product. It is important to note here that the size of the list that you encounter in letter-saving, like the size of this L here, is exponential in the dimension of your space, and for the information set decoding setting, the metric space would be the hamming cube with the hamming distance, the L1 distance, and again the size of the list would be exponential in the dimension of your space, and this look at the sense of hashing has been studied outside cryptography for a long time already, and people have been studying this for many different metrics and such, but it is important to note that in a nearest neighbor literature, commonly people consider the case where the size of the list is sub-exonential in the dimension, so people have also been looking at lower bounds for this hatch-based model, for look at sense of hashing in general, but they are usually in this case where the size of the list is sub-exonential in the dimension, and that means that they do not apply to the setting here for letter-saving or information decoding, because we have an exponential side of the list. Okay, so these hash functions, like ideally we would find such hash functions where the gap between the left-hand side and the right-hand side is as big as possible. Okay, so just to instantiate this on the Euclidean sphere, then you can just replace this M by the Unisphere, and two vectors being close, you can also interpret this as the dot product between X and Y being larger than some parameter gamma in this case, and note that the hash of X is equal to the hash of Y if they are both equal to some value N, and you can sum over all of these N to rewrite these probabilities as a summation in this case, like this on both the left-hand side and the right-hand side, and notice that the hash of X is equal to N if X is in the preimage of N under the hash function, so you can imagine that this hash function, like there's some kind of partition of the sphere where each part of this partition has a different hash value, and the H inverse of N is like some region of the sphere which maps under H to this hash value, and so the hash of X being equal to the hash of Y being equal to N can be rewritten as that X and Y are both in the preimage of N. Usually, these hash regions have similar shapes for all values and like for all hash values. You cannot really assume that if you want to prove a lower bound that for this talk, if you just assume that for now, just for intuition, then you can see that you can rewrite these sums as just N times one of these terms, say the H inverse of zero. Then again, to have the left side much bigger than the right side, then we can also just eliminate the N on both sides now. And now we can also rewrite the H inverse of zero as some region A on the sphere. And note that the right-hand side, this is a probability that X and Y are in this region A, where X and Y are independent and uniformly random from the sphere. So we can rewrite the right-hand side as the surface area, the relative volume of A compared to the unisphere. And we want to find functions for which these partitions or for which these regions A on the left-hand side here are much bigger than the relative volume squares of this region A. So completely, you can rephrase this whole problem, roughly speaking, as that if you are given a fixed size of your region on the sphere. So if you are given that the size of this A has to be like a quarter of the sphere, say you want to find the region which maximizes the left-hand side, which maximizes the probability that if two vectors are actually close, then they are both in the same region with the maximum probability that you can get for this size frame. So actually to get a lower bound here, what we used is the Bernstein-Taylor inequality from the 70s from the last century. This is kind of abstract formal statements. But yeah, what you have here is some inequality between left-hand sides, double integral of F, G and H.  Like F star is a function of F in a sense. Like if you are given F, then you can construct the unique F star, which has this property that it is a symmetric non-decreasing rearrangement of F and the same for G. So okay, actually we just use this by, we use this for a specific choice of F, G and H. In this case, F of X being the indicated function of X being in some set A. We have not fixed A yet, but for any A we can use these F and G. And the H here is defined as the indicator of that a dot product is bigger than gamma. And if you use these F and G, then actually the F star and the G star that you get are the indicated functions that X and Y are in C of A. Where C of A is the spherical cap with the same volume as A. So like the sigma of A is the same as sigma of C of A. So that's just the F star that you get if you take this F. That's just what you get. And you can plug this into this inequality that has been proven before. So then you get the left-hand side and the right-hand side here. And now you can look, if you look closely at what it really says on the left-hand side and the right-hand side, then you can see that these are really just probabilities over the unisphere. And actually the left-hand side is just the probability over X and Y uniformly at random from the sphere that X and Y are in this region A. And that the dot product is larger than gamma. And the same on the right-hand side except that you don't have A, but you have this spherical cap with the same volume as A. And then, okay, now you have a probability over X and Y being A and X dot A being bigger than gamma. But you can divide by the probability that X dot Y is bigger than gamma and then you just get this inequality from that. Which is kind of like what we were looking for on the previous slide. Like we wanted to maximize this probability that X and Y are in A, given that X and Y are uniformly at random from the sphere condition from X and Y having a dot product larger than gamma. And now we have this inequality which tells you that, actually this one, this probability is maximized when you take the spherical cap with the same volume as A. And so, like, if you draw it in the figure, then you can see like the, if you want X and Y to be often in the same region A, where X and Y are close on the sphere, then the best you can do is like use these spherical caps on the right. Like this wiggly shape A is not really going to be effective. The best you can really do is use spherical caps. So I'll just go back to the previous slide, like this probability that we have here. This inequality, this Mercy-Taylor inequality tells us that this probability is always at most equal to the right-hand side. And we wanted to maximize it. So we really want to make this an equality. And we can make this an equality because this C of A on the right is actually has the same volume as A. So like this has the same, yeah, got this volume on the sphere. And so this really tells us that the solution to this problem here is to use spherical caps. And note that, yeah, okay, if you now want to go back to proving that these high-scratches are optimal, for instance, you also need to take into account that maybe the shapes of the regions are not exactly the same. Yeah, and you need to have this sum over different regions as well. So I mean, there's more details in the paper to go through this proof. But like this is the main intuition that tells you like, okay, the actual, the best shape that you can use is the spherical cap. And it comes from this, it's not from Mercy-Taylor from the 70s, yeah, from the 70s. And so, okay, so if you then use this result, so ultimately, when you complete the proof, then you get this conditional bound that if you do lattice-saving with hash-based nearest-neighbour-searching, then you cannot do better than using these spherical cap shapes. And this has already been studied by in many different settings like for classical saving, for instance, like which is like just combining pairs of factors using some hash-based approach. Then, yeah, now we know that this result from 2016 with this 292 exponent is actually conditionally optimal, conditional on being in this hash-based setting. You can also do saving and then do some quantum techniques on top of it, for instance, grover-searching. So this has been studied in 2016 as well. And then we're shown that you get this exponent 0.265, actually better than the 90. And again, this is now conditionally optimal, conditional on using lattice-saving with some form of hash-based searching and using grover-searching specifically. Like maybe you can do better with all the quantum techniques and actually recently there was this result from 2021, which showed that if you do lattice-saving again with some form of nearest-neighbour-searching, but then you don't use grover, but you use quantum random walks and actually you get an even better exponent like 0.2570. But this improves the quantum part. It does not improve the nearest-neighbour part. So this does not actually violate the lower bound that we proved slightly before this recently. Everything is fine. And also for tuple-saving or for closest-vector algorithms, again, these results are now conditionally optimal, conditional on that you use some form of hash-based nearest-neighbour-searching. And again, there may be different ways of doing nearest-neighbour-searching or closest-vector searching, but like so far, at least in lattice-saving, people have really studied hash-based approaches a lot like several of these shifts that have appeared in the last decade, they're all based on hash-based nearest-neighbour-searching. Okay, and then just to conclude with a few open problems that might be of interest. So, yeah, as I stated before, so all these results are conditional on that you are using some form of hash-based nearest-neighbour-searching to solve the closest pairs problem. But yeah, maybe you can actually use some other closest pair techniques, which actually don't fall within this model, and then you can still get better results. I mean, it might be possible, so who knows? I mean, you're just not going to do better within this hash-based model. Also, this only affects the closest pairs subroutine of these algorithms. But yeah, you may be able to improve other pairs like people have been doing with, say, this Jessica, for instance, for letter sieving. You just improve the base reduction or you improve some sub-extrational overhead somewhere. I mean, that's still worth to do there. Also, these are only, like, asymptotics about the leading constant. So these results are only saying that asymptotically, if you are only focused on minimizing the constant in the exponent, then you are not going to do better than, like, using these spherical caps for letter sieving, at least. But maybe you can decrease the sub-extrational overhead like either using the same method or even using different methods which have a worse constant in the exponent but have a lower overhead that might already be better. So because, I mean, concretely, we are interested in parameters that we're using in cryptography and not just in asymptotics. And also the bound for ISD. I didn't really go into it in this talk, but the bound is not tight, as I mentioned before. So either there may be better techniques to get better upper bounds or there may be better lower bounds which actually do match the correct techniques. So that is still also an open problem and we haven't solved it yet. Anyway, so, yeah, thank you for watching.


Question 47, Part 1 of Summa Theologica Secunda Secunde, Triatis on the Cardinal Virtues, The Virtue of Prudence. This is a LibriVox recording. All LibriVox recordings are in the public domain. For more information or to volunteer, please visit LibriVox.org. Summa Theologica Secunda Secunde, Triatis on the Cardinal Virtues, The Virtue of Prudence by St. Thomas Aquinas, translated by the Fathers of the English Dominican Province. Question 47 of Prudence, considered in itself in 16 articles. Part 1, Articles 1 through 5. After treating of the theological virtues, we must in due sequence consider the cardinal virtues. In the first place, we shall consider prudence in itself. Secondly, its parts. Thirdly, the corresponding gift. Fourthly, the contrary vices. Fifthly, the precepts concerning prudence. Under the first head, there are 16 points of inquiry. First, whether prudence is in the will or in the reason. Second, if in the reason, whether it is only in the practical or also in the speculative reason. Third, whether it takes cognizance of singulars. Fourth, whether it is virtue. Fifth, whether it is a special virtue. Sixth, whether it appoints the end to the moral virtues. Seventh, whether it fixes the mean in the moral virtues. Eighth, whether its proper act is command. Ninth, whether solicitude or watchfulness belongs to prudence. Tenth, whether prudence extends to the governing of many. Eleventh, whether the prudence which regards private good is the same in species as that which regards the common good. Twelfth, whether prudence is in subjects or only in their rulers. Thirteenth, whether prudence is in the wicked. Fourteenth, whether prudence is in all good men. Fifteenth, whether prudence is in us naturally. Sixteenth, whether prudence is lost by forgetfulness. First article, whether prudence is in the cognitive or in the appetitive faculty. Objection one, you would see that prudence is not in the cognitive but in the appetitive faculty. For Augustine says, on the morals of the Catholic Church, 15, prudence is love choosing wisely between the things that help and those that hinder. Now love is not in the cognitive but in the appetitive faculty. Therefore prudence is in the appetitive faculty. Objection two, further, as appears from the foregoing definition it belongs to prudence to choose wisely. But choice is an act of the appetitive faculty as stated above. Paras Prima Sukundey, question 13, article one. Therefore prudence is not in the cognitive but in the appetitive faculty. Objection three, further, the philosopher says in ethics six five that in art it is better to err voluntarily than involuntarily whereas in the case of prudence as of the virtues it is worse. Now the moral virtues of which he is treating there are in the appetitive faculty whereas art is in the reason. Therefore prudence is in the appetitive rather than in the rational faculty. On the contrary Augustine says in his 83 questions, question 61, prudence is the knowledge of what to seek and what to avoid. I answer that as Isidor says in his etymologies 10, a prudent man is one who sees as it were from afar for his sight is keen and he foresees the event of uncertainties. Now sight belongs not to the appetitive but to the cognitive faculty wherefore it is manifest that prudence belongs directly to the cognitive and not to the sensitive faculty because by the latter we know nothing but what is within reach and offers itself to the senses. While to obtain knowledge of the future from knowledge of the present or past which pertains to prudence belongs properly to the reason because this is done by a process of comparison. It follows therefore that prudence properly speaking is in the reason. Reply to objection one. As stated above in Paras Prima, question 82, article 4, the will moves all the faculties to their acts. Now the first act of the appetitive faculty is love as stated above. Paras Prima Sekunde, question 25, articles 1 and 2. Accordingly prudence is said to be love not indeed essentially but insofar as love moves to the act of prudence. Wherefore Augustine goes on to say that prudence is love discerning a right, that which helps from that which hinders us intending to God. Now love is said to discern because it moves the reason to discern. Reply to objection two. The prudent man considers things afar off insofar as they tend to be a help or a hindrance to that which has to be done at the present time. And it is clear that those things which prudence considers stand in relation to this other as in relation to the end. Now of those things that are directed to the end there is counsel in the reason and choice in the appetite of which to counsel belongs more properly to prudence since the philosopher states in ethics 6, 5, 7 and 9 that a prudent man takes good counsel. But as choice presupposes counsel since it is the desire for what has been already counseled, ethics 3, 2. It follows that choice can also be ascribed to prudence indirectly insofar to it as prudence directs the choice by means of counsel. Reply to objection three. The worth of prudence consists not in thought merely but in its application to action which is the end of the practical reason. Wherefore if any defect occur in this it is most contrary to prudence since the end being of most import in everything it follows that a defect which touches the end is the worst of all. Hence the philosopher goes on to say in ethics 6, 5 that prudence is something more than a merely rational habit such as art is since as stated above in Parse Prima Secunde question 57 article 4. It includes application to action which application is an act of the will. Second article. Whether prudence belongs to the practical reason alone or also to the speculative reason. Objection one. You would seem that prudence belongs not only to the practical but also to the speculative reason. For it is written in Proverbs 10 23. Wisdom is prudence to a man. Now wisdom consists chiefly in contemplation therefore prudence does also. Objection two. Further Ambrose says on the duties of the clergy 124 prudence is concerned with the quest of truth and fills us with a desire of fuller knowledge. Now this belongs to the speculative reason. Therefore prudence resides also in the speculative reason. Objection three. Further the philosopher assigns art and prudence to the same part of the soul. Ethics 6 1. Now art may be not only practical but also speculative as in the case of the liberal arts. Therefore prudence also is both practical and speculative. On the contrary the philosopher says in Ethics 6 5 that prudence is right reason applied to action. Now this belongs to none but the practical reason. Therefore prudence is in the practical reason only. I answer that according to the philosopher in Ethics 6 5 a prudent man is one who is capable of taking good counsel. Now counsel is about things that we have to do in relation to some end and the reason that deals with things to be done for an end is the practical reason. Hence it is evident that prudence resides only in the practical reason. Reply to Objection one. As stated above in Question 45 Articles 1 and 3. Wisdom considers the absolutely highest cause so that the consideration of the highest cause in any particular genus belongs to wisdom in that genus. Now in the genus of human acts the highest cause is the common end of all life and it is this end that prudence intends. For the philosopher says in Ethics 6 5 that just as he who reasons well for the realization of a particular end such as victory is said to be prudent not absolutely but in a particular genus namely warfare so that he reasons well with regard to write conduct as a whole is said to be prudent absolutely. Therefore it is clear that prudence is wisdom about human affairs but not wisdom absolutely because it is not about the absolutely highest cause for it is about human good and this is not the best thing of all and so it is stated significantly that prudence is wisdom for man but not wisdom absolutely. Reply to Objection two. Ambrose and Tully also in Rhetorix 253. Take the word prudence in a broad sense for any human knowledge whether speculative or practical and yet it may also be replied that the act itself of the speculative reason insofar as it is voluntary is a matter of choice and counsel as to its exercise and consequently comes under the direction of prudence on the other hand as regards its specification in relation to its object which is the necessary true it comes under neither counsel nor prudence. Reply to Objection three. Every application of right reason in the work of production belongs to art but to prudence belongs only the application of right reason in matters of counsel which are those wherein there is no fixed way of obtaining the end as stated in Ethics 33. Since then the speculative reason makes things such as syllogisms, propositions and the like wherein the process follows certain and fixed rules consequently in respect of such things it is possible to have the essentials of art but not of prudence and so we find such a thing as speculative art but not a speculative prudence. Third article. Whether prudence takes cognizance of singulars. Objection one. It would seem that prudence does not take cognizance of singulars for prudence is in the reason as stated above in articles one and two but reason deals with universals according to physics one five therefore prudence does not take cognizance except of universals. Objection two. Further, singulars are infinite in number but the reason cannot comprehend an infinite number of things therefore prudence which is right reason is not about singulars. Objection three. Further, particulars are known by the senses but prudence is not a sense for many persons who have keen outward senses are devoid of prudence therefore prudence does not take cognizance of singulars on the contrary the philosopher says in ethics six seven that prudence does not deal with universals only but needs to take cognizance of singulars also I answer that as stated above in articles one third reply to prudence belongs not only the consideration of the reason but also the application to action which is the end of the practical reason but no man can conveniently apply one thing to another unless he knows both the thing to be applied and the thing to which it has to be applied now actions are in singular matters and so it is necessary for the prudent man to know both the universal principles of reason and the singulars about which actions are concerned reply to objection one reason first and chiefly is concerned with universals and yet it is able to apply universal rules to particular cases hence the conclusions of syllogisms are not only universal but also particular because the intellect by a kind of reflection extends to matter as stated in on the soul number three reply to objection two it is because the infinite number of singulars cannot be comprehended by human reason that our councils are uncertain wisdom nine fourteen nevertheless experience reduces the infinity of singulars to a certain finite number which occur as a general rule and the knowledge of these suffices for human prudence reply to objection three as the philosopher says in ethics six eight prudence does not reside in the external senses whereby we know sensible objects but in the interior sense which is perfected by memory and experience so as to judge promptly of particular cases this does not mean however that prudence is in the interior sense as in its principal subject for it is chiefly in the reason yet by a kind of application it extends to this sense fourth article whether prudence is a virtue objection one it would seem that prudence is not a virtue for gustin says on the free choice of the will one thirteen that prudence is the science of what to desire and what to avoid now science is condivided with virtue as appears in the categories six therefore prudence is not a virtue objection two further there is no virtue of a virtue but there is a virtue of art as the philosopher states in ethics six five where for art is not a virtue now there is prudence in art for it is written in second chronicles two fourteen concerning herum that he knew to grave all sort of graving and to devise ingeniously prudenter all that there may be need of in the work therefore prudence is not a virtue objection three further no virtue can be immoderate but prudence is a moderate else it would be useless to say as in proverbs twenty three four set bounds to thy prudence therefore prudence is not a virtue on the contrary gregory states on the morals of jobe two forty nine that prudence temperance fortitude and justice are four virtues i answer that as stated above in bar's prima secunde question 55 article three and question 56 article one when we were treating a virtues in general virtue is that which makes its possessor good and his work good likewise now good may be understood in a twofold sense first materially for the thing that is good secondly formally under the aspect of good good under the aspect of good is the object of the appetitive power hence if any habits rectify the consideration of reason without regarding the rectitude of the appetite they have less of the nature of a virtue since they direct man to good materially that is to say to the thing which is good but without considering it under the aspect of good on the other hand those virtues which regard the rectitude of the appetite have more of the nature of virtue because they consider the good not only materially but also formally in other words they consider that which is good under the aspect of good now it belongs to prudence as stated above in articles one third apply article three to apply right reason to action and this is not done without a right appetite hence prudence as the nature of virtue not only as the other intellectual virtues have it but also as the moral virtues have it among which virtues it is enumerated reply to objection one august and there take science in the broad sense for any kind of right reason reply to objection two the philosopher says that there is a virtue of art because art does not require rectitude of the appetite therefore in order that a man may make right use of his art he needs to have a virtue which will rectify his appetite prudence however has nothing to do with the matter of art because art is both directed to a particular end and has fixed means of obtaining that end and yet by a kind of comparison a man may be said to act prudently in matters of art moreover in certain arts on account of the uncertainty of the means for obtaining the end there is need for counsel as for instance in the arts of medicine and navigation as stated in ethics three three reply to objection three this saying of the wise man does not mean that prudence itself should be moderate but that moderation must be imposed on other things according to prudence fifth article whether prudence is a special virtue objection one it would seem that prudence is not a special virtue for no special virtue is included in the definition of virtue in general since virtue is defined in ethics two six an elective habit that follows a mean appointed by reason in relation to ourselves even as a wise man decides now right reason is reason in accordance with prudence as stated in ethics six thirteen therefore prudence is not a special virtue objection two further the philosopher says in ethics six thirteen that the effect of moral virtue is right action as regards the end and that of prudence right action as regards the means now in every virtue certain things have to be done as a means to the end therefore prudence is in every virtue and consequently it is not a special virtue objection three further a special virtue has a special object but prudence has not a special object for it is right reason applied to action as stated in ethics six five and all works of virtue are actions therefore prudence is not a special virtue on the contrary it is distinct from and numbered among the other virtues for it is written in wisdom eight seven she teaches temperance and prudence justice and fortitude i answer that since acts and habits take their species from their objects as shown above paris primus akundi question one article three question eighteen article two question fifty four article two any habit that has a corresponding special object distinct from other objects must needs be a special habit and if it be a good habit it must be a special virtue now an object is called special not merely according to the consideration of its matter but rather according to its formal aspect as explained above paris primus akundi question 54 article two first reply because one and the same thing is the subject matter of the acts of different habits and also of different powers according to its different formal aspects now a yet greater difference of object is requisite for a difference of powers and for a difference of habits since several habits are found in the same power as stated above paris primus akundi question 54 article one consequently any difference in the aspect of an object that requires a difference of powers will a fourth cri require a difference of habits accordingly we must say that since prudence is in the reason as stated above article two it is differentiated from the other intellectual virtues by a material difference of objects wisdom knowledge and understanding are about necessary things whereas art and prudence are about contingent things art being concerned with things made that is with things produced in external matter such as a house a knife and so forth and prudence being concerned with things done that is with things that have their being in the doer himself as stated above paris primus akundi question 57 article four on the other hand prudence is differentiated from the moral virtues according to a formal aspect distinctive of powers that is the intellect of power wherein is prudence and the appetitive power wherein is moral virtue hence it is evident that prudence is a special virtue distinct from all other virtues reply to objection one this is not a definition of virtue in general but of moral virtue the definition of which fittingly includes an intellectual virtue notably prudence which has the same matter in common with moral virtue because just as the subject of moral virtue is something that partakes of reason so moral virtue has the aspect of virtue insofar as it partakes of intellectual virtue reply to objection two this argument proves that prudence helps all the virtues and works in all of them but this does not suffice to prove that it is not a special virtue for nothing prevents a certain genus from containing a species which is operative in every other species of that same genus even as the sun has an influence over all bodies reply to objection three things done are indeed the matter of prudence insofar as they are the object of reason that is considered as true but they are the matter of the moral virtues insofar as they are the object of the appetitive power that is considered as good end of question 47 part 1 read by Michael Shane Greg Lambert LC question 47 part 2 of Summa Theologica Secunda Secunde triates on the cardinal virtues the virtue of prudence this is a LibriVox recording all LibriVox recordings are in the public domain for more information or to volunteer please visit LibriVox.org Summa Theologica Secunda Secunde triates on the cardinal virtues the virtue of prudence by St. Thomas Aquinas translated by the fathers of the English Dominican province question 47 of prudence considered in itself in 16 articles part 2 articles 6 through 10 sixth article whether prudence appoints the end to moral virtues objection 1 it would seem that prudence appoints the end to moral virtues since prudence is in the reason while moral virtue is in the appetite it seems that prudence stands in relation to moral virtue as reason to the appetite now reason appoints the end to the appetitive power therefore prudence appoints the end to the moral virtues objection 2 further man surpasses irrational beings by his reason but he has other things in common with them accordingly the other parts of man are in relation to his reason what man is in relation to irrational creatures now man is the end of irrational creatures according to politics 1 3 therefore all the other parts of man are directed to reason as to their end but prudence is right reason applied to action as stated above in article 2 therefore all actions are directed to prudence as their end therefore prudence appoints the end to all moral virtues objection 3 further it belongs to the virtue art or power that is concerned about the end to command the virtues or arts that are concerned about the means now prudence disposes of the other moral virtues and commands them therefore it appoints their end to them on the contrary the philosopher says in ethics 612 that moral virtue ensures the rectitude of the intention of the end while prudence ensures the rectitude of the means therefore it does not belong to prudence to appoint the end to moral virtues but only to regulate the means i answer that the end of moral virtues is human good now the good of the human soul is to be in accord with reason as Dionysius declares on the divine names 4 wherefore the ends of moral virtue must have necessity pre-exist in the reason now just as in the speculative reason there are certain things naturally known about which is understanding and certain things of which we obtain knowledge through them notably conclusions about which is science so in the practical reason certain things pre-exist as naturally known principles and such are the ends of the moral virtues since the end is in practical matters what principles are in speculative matters as stated above question 23 article 7 second reply parts prima secunde question 13 article 3 while certain things are in the practical reason by way of conclusions and such are the means which we gather from the ends themselves about these is prudence which applies universal principles to the particular conclusions of practical matters consequently it does not belong to prudence to appoint the end to moral virtues but only to regulate the means reply to objection one natural reason known by the name of cindersis appoints the end to moral virtues as stated above parts prima question 79 article 12 but prudence does not do this for the reason given above this suffices for the reply to the second objection reply to objection three the end concerns the moral virtues not as though they appointed the end but because they tend to the end which is appointed by natural reason in this they are helped by prudence which prepares the way for them by disposing the means and said follows that prudence is more excellent than the moral virtues and moves them yet cindersis moves prudence just as the understanding of principles move science seventh article whether it belongs to prudence to find the mean in moral virtues objection one you would seem that it does not belong to prudence to find the mean in moral virtues for the achievement of the mean is the end of moral virtues but prudence does not appoint the end to moral virtues as shown above in article six therefore it does not find the mean in them objection two further that which of itself as being would seem to have no cause but its very being is its cause since a thing is said to have being by reason of its cause now to follow the mean belongs to moral virtue by reason of itself as part of its definition is shown above article five objection one therefore prudence does not cause the mean in moral virtues objection three further prudence works after the manner of reason but moral virtue tends to the mean after the manner of nature because as tully states in rhetoric's 253 virtue is a habit like a second nature in accord with reason therefore prudence does not appoint the mean to moral virtues on the contrary in the foregoing definition of moral virtue article five objection one it is stated that it follows a mean appointed by reason even as a wise man decides i answer that the proper end of each moral virtue consists precisely in conformity with right reason for temperance intends that man should not stray from reason for the sake of his concupiscences fortitude that he should not stray from right judgment of reason through fear or daring moreover this end is appointed to man according to natural reason since natural reason dictates to each one that he should act according to reason but it belongs to the ruling of prudence to decide in what manner and by what means man shall obtain the mean of reason in his deeds for though the attainment of the mean is the end of a moral virtue yet this mean is found by the right disposition of these things that are directed to the end this suffices for the reply to the first objection reply to objection two just as a natural agent makes a form to be in matter yet does not make that which is essential to the form to belong to it so too prudence appoints the mean in passions and operations and yet does not make the searching of the mean to belong to virtue reply to objection three moral virtue after the manner of nature intends to attain the mean since however the mean as such is not found in all matters after the same manner it follows that the inclination of nature which ever works in the same manner does not suffice for this purpose and so the ruling of prudence is required eighth article whether command is the chief act of prudence objection one it would seem that command is not the chief act of prudence for command regards the good to be ensued now agustin in on the trinity 149 states that it is an act of prudence to avoid ambushes therefore command is not the chief act of prudence objection two further the philosopher says in ethics 65 that the prudent man takes good counsel now to take counsel and to command seem to be different acts as appears from what has been said above bars prima secunde question 57 article six therefore command is not the chief act of prudence objection three further it seems to belong to the will to command and to rule since the will has the end for its object and moves the other powers of the soul now prudence is not in the will but in the reason therefore command is not an act of prudence on the contrary the philosopher says in ethics 610 that prudence commands i answer that prudence is right reason applied to action as stated above in article two hence that which is the chief act of reason in regard to action must needs be the chief act of prudence now there are three such acts the first is to take counsel which belongs to discovery for counsel is an act of inquiry as stated above bars prima secunde question 14 article one the second act is to judge of what one has discovered and this is an act of the speculative reason but the practical reason which is directed to action goes further and its third act is to command which act consists in applying to action the things counseled and judged and since this act approaches nearer to the end of the practical reason it follows that it is the chief act of the practical reason and consequently of prudence in confirmation of this we find that the perfection of art consists in judging and not in commanding wherefore he who sins voluntarily against his craft is reputed a better craftsman than he who does so involuntarily because the former seems to do so from right judgment and the latter from a defective judgment on the other hand it is the reverse in prudence as stated in ethics six five for it is more imprudent to sin voluntarily since this is to be lacking in the chief act of prudence notably command than to sin involuntarily reply to objection one the act of command extends both to the ensuing of good and to the avoidance of evil nevertheless augustin obscribes the avoidance of ambushes to prudence not as its chief act but as an act of prudence that does not continue in heaven reply to objection two good counsel is required in order that the good things discovered may be applied to action wherefore command belongs to prudence which takes good counsel reply to objection three simply to move belongs to the will but command denotes motion together with a kind of ordering wherefore it is an act of the reason as stated above bars premissa kunde question 17 article one ninth article whether solicitude belongs to prudence objection one he would seem that solicitude does not belong to prudence for solicitude implies disquiet wherefore is it or says in his etymologies 10 that a solicitous man is a restless man now motion belongs chiefly to the appetitive power wherefore solicitude does also but prudence is not in the appetitive power but in the reason as stated above in article one therefore solicitude does not belong to prudence objection two further the certainty of truth seems opposed to solicitude wherefore it is related in first king's 920 that samuel said to sol as for the asses which were lost three days ago be not solicitous because they are found now the certainty of truth belongs to prudence since it is an intellectual virtue therefore solicitude is in opposition to prudence rather than belonging to it objection three further the philosopher says in ethics for three that the magnanimous man is slow and leisurely now slowness is contrary to solicitude since then prudence is not opposed to magnanimity for good is not opposed to good as stated in the categories number eight it would seem that solicitude does not belong to prudence on the contrary it is written in first peter four seven be prudent and watch in prayers but watchfulness is the same as solicitude therefore solicitude belongs to prudence i answer that according to isidor etymologies ten a man is said to be solicitous through being shrewd solers and alert cheetus insofar as a man through a certain shrewdness of mind is on the alert to do whatever has to be done now this belongs to prudence whose chief act is a command about what has been already counseled and judged in matters of action hence the philosopher says in ethics six nine that one should be quick in carrying out the counsel taken but slow in taking counsel hence it is that solicitude belongs properly to prudence and for this reason augustin says on the morals of the catholic church 24 that prudence keeps most careful watch and word lest by degrees we be deceived unawares by evil counsel reply to objection one movement belongs to the appetitive power as to the principle of movement in accordance however with the direction and command of reason where in solicitude consists reply to objection two according to the philosopher in ethics one three equal certainty should not be sought in all things but in each matter according to its proper mode and since the matter of prudence is the contingent singulars about which are human actions the certainty of prudence cannot be so great as to be devoid of all solicitude reply to objection three the magnanimous man is said to be slow and leisurely not because he is solicitous about nothing but because he is not over solicitous about many things and is trustful in matters where he ought to have trust and is not over solicitous about them for over much fear and distrust are the cause of over solicitude since fear makes us take counsel as stated above parts prima secundae question 44 article 2 when we were treating of the passion of fear 10th article whether prudence extends to the governing of many objection one it would seem that prudence does not extend to the governing of many but only to the government of oneself for the philosopher says in ethics five one that virtue directed to the common good is justice but prudence differs from justice therefore prudence is not directed to the common good objection to further he seems to be prudent who seeks and does good for himself now those who seek the common good often neglect their own therefore they are not prudent objection three further prudence is specifically distinct from temperance and fortitude but temperance and fortitude seem to be related only to a man's own good therefore the same applies to prudence on the contrary our lord said matthew 2445 who thinkest thou is a faithful and prudent servant whom his lord hath appointed over his family note the duly bible translates prudent as wise i answer that according to the philosopher in ethics six eight some have held that prudence does not extend to the common good but only to the good of the individual and this because they thought that man is not bound to seek other than his own good but this opinion is opposed to charity which seeketh not her own first Corinthians 13 5 where for the apostle says of himself in first Corinthians 10 33 not seeking that which is profitable to myself but to many that they may be saved moreover it is contrary to right reason which judges the common good to be better than the good of the individual accordingly since it belongs to prudence rightly to counsel judge and command concerning the means of obtaining a due end it is evident that prudence regards not only the private good of the individual but also the common good of the multitude reply to objection one the philosopher is speaking there of moral virtue now justice every moral virtue that is directed to the common good is called legal justice so the prudence that is directed to the common good is called political prudence for the latter stands in the same relation to legal justice as prudence simply so called to moral virtue reply to objection two he that seeks the good of the many seeks in consequence his own good for two reasons first because the individual good is impossible without the common good of the family state or kingdom hence Valerius maximus says in his nine books of memorable deeds and sayings four six of the ancient romans that they would rather be poor in a rich empire than rich in a poor empire secondly because since man is a part of the home and state he must needs consider what is good for him by being prudent about the good of the many for the good disposition of parts depends on their relation to the whole thus Augustine says in his confessions three eight that any part which does not harmonize with its whole is offensive reply to objection three even temperance and fortitude can be directed to the common good hence there are precepts of law concerning them as stated in ethics five one more so however prudence and justice since these belong to the rational faculty which directly regards the universal just as the sensitive part regards singular's end of question forty seven part two read by Michael Shane Craig Lambert LC question forty seven part three of Summa Theologica Secunda Secunde triates on the cardinal virtues the virtue of prudence this is a LibriVox recording all LibriVox recordings are in the public domain for more information or to volunteer please visit LibriVox.org Summa Theologica Secunda Secunde triates on the cardinal virtues the virtue of prudence by Saint Thomas Aquinas translated by the fathers of the English Dominican province question forty seven of prudence considered in itself in 16 articles part three articles 11 through 16 11th article whether prudence about one's own good is specifically the same as that which extends to the common good objection one it seems that prudence about one's own good is the same specifically is that which extends to the common good for the philosopher says in ethics six eight that political prudence and prudence are the same habit yet their essence is not the same objection two further the philosopher says in politics three two that virtue is the same in a good man and in a good ruler now political prudence is chiefly in the ruler in whom it is architectonic as it were since then prudence is a virtue of a good man it seems that prudence and political prudence are the same habit objection three further a habit is not diversified in species or essence by things which are subordinate to one another but the particular good which belongs to prudence simply so-called is subordinate to the common good which belongs to political prudence therefore prudence and political prudence differ neither specifically nor essentially on the contrary political prudence which is directed to the common good of the state domestic economy which is of such things as relate to the common good of the household or family and monastic economy which is concerned with things affecting the good of one person are all distinct sciences therefore in like manner there are three kinds of prudence corresponding to the above differences of matter i answer that as stated above in article five as well in question 54 article two first reply the species of habits differ according to the difference of object considered in its formal aspect now the formal aspect of all things directed to the end is taken from the end itself as shown above bars prima secunde prologue as well as question 102 article one where for the species of habits differ by the relation to different ends again the individual good the good of the family and the good of the city and kingdom are different ends where for there must needs be different species of prudence corresponding to these different ends so that one is prudence simply so-called which is directed to one's own good another domestic prudence which is directed to the common good of the home and a third political prudence which is directed to the common good of the state or kingdom reply to objection one the philosopher means not that political prudence is substantially the same habit as any kind of prudence but that it is the same as the prudence which is directed to the common good this is called prudence in respect of the common notion of prudence that is as being right reason applied to action while it is called political as being directed to the common good reply to objection two as the philosopher declares in politics three two it belongs to a good man to be able to rule well and to obey well where for the virtue of a good man includes also that of a good ruler yet the virtue of the ruler and of the subject differs specifically even as the virtue of a man and of a woman as stated by the same authority politics three two reply to objection three even different ends one of which is subordinate to the other diversify the species of a habit thus for instance habits directed to riding soldiering and civic life differ specifically although their ends are subordinate to one another in like manner though the good of the individual is subordinate to the good of the many that does not prevent this difference from making the habits differ specifically but it follows that the habit which is directed to the last end is above the other habits and commands them twelfth article whether prudence is in subjects or only in their rulers objection one you would seem that prudence is not in subjects but only in their rulers for the philosopher says in politics three two that prudence alone is the virtue proper to a ruler while other virtues are common to subjects and rulers and the prudence of the subject is not a virtue but a true opinion objection two further it is stated in politics one five that a slave is not competent to take counsel but prudence makes a man take good counsel ethics six five therefore prudence is not defeating slaves or subjects objection three further prudence exercises command is stated above in article eight but command is not in the competency of slaves or subjects but only of rulers therefore prudence is not in subjects but only in rulers on the contrary the philosopher says in ethics six eight that there are two kinds of political prudence one of which is legislative and belongs to rulers while the other retains the common name political and is about individual actions now it belongs also to subjects to perform these individual actions therefore prudence is not only in rulers but also in subjects I answer that prudence is in the reason now ruling and governing belong properly to the reason and therefore it is proper to a man to reason and be prudent insofar as he has a share in ruling and governing but it is evident that the subject as subject and the slave as slave are not competent to rule and govern but rather to be ruled and governed therefore prudence is not in the virtue of a slave as slave nor of a subject as subject since however every man for as much as he is rational has a share in ruling according to the judgment of reason he is proportionately competent to have prudence therefore it is manifest that prudence is in the ruler after the manner of a mastercraft ethics six eight but in the subjects after the manner of a handicraft replied to objection one the saying of the philosopher is to be understood strictly namely that prudence is not the virtue of a subject as such replied to objection two a slave is not capable of taking counsel insofar as he is a slave for thus he is the instrument of his master but he does take counsel insofar as he is a rational animal replied to objection three by prudence a man commands not only others but also himself insofar as the reason is said to command the lower powers thirteenth article whether prudence can be in sinners objection one it would seem that there can be prudence in sinners for our lord said in luke sixteen eight the children of this world are more prudent in their generation than the children of light note the dewey translation translates prudent as wiser now the children of this world are sinners therefore there be prudence in sinners objection two further faith is a more excellent virtue than prudence but there can be faith in sinners therefore there can be prudence also objection three further according to ethics six seven we say that to be of good counsel is the work of prudent man especially now many sinners can take good counsel therefore sinners can have prudence on the contrary the philosopher declares in ethics six twelve that it is impossible for a man to be prudent unless he be good now no sinner is a good man therefore no sinner is prudent i answer that prudence is threefold there is a false prudence which takes its name from its likeness to true prudence for since a prudent man is one who disposes well of the things that have to be done for a good end whoever disposes well of such things as are fitting for an evil end has false prudence insofar as that which he takes for an end is good not in truth but in appearance thus man is called a good robber and in this way we may speak of a prudent robber by way of similarity because he devises fitting ways of committing robbery this is the prudence of which the apostle says in romans eight six the prudence of the flesh is death note the dewey translation translates prudence as wisdom because to wit it places its ultimate end in the pleasures of the flesh the second prudence is indeed true prudence because it devises fitting ways of obtaining a good end and yet it is imperfect from a twofold source first because the good which it takes for an end is not the common end of all human life but of some particular affair thus when a man devises fitting ways of conducting business or of sailing a ship he is called a prudent businessman or a prudent sailor secondly because he fails in the chief act of prudence as when a man takes counsel a right and forms a good judgment even about things concerning life as a whole but fails to make an effective command the third prudence is both true and perfect for it takes counsel judges and commands a right in respect of the good end of man's whole life and this alone is prudence simply so called and cannot be in sinners whereas the first prudence is in sinners alone while imperfect prudence is common to good and wicked men especially that which is imperfect through being directed to a particular end since that which is imperfect on account of a failing in the chief act is only in the wicked reply to objection one this saying of our lord is to be understood of the first prudence therefore it is not said that they are prudent absolutely but that they are prudent in their generation reply to objection two the nature of faith consists not in conformity with the appetite for certain right actions but in knowledge alone on the other hand prudence implies a relation to a right appetite first because its principles are the ends in matters of action and of such ends one forms a right estimate through the habits of moral virtue which rectify the appetite therefore without the moral virtues there is no prudence as shown above bars prima secunde question 58 article 5 secondly because prudence commands right actions which does not happen unless the appetite be right therefore through faith on account of its object is more excellent than prudence yet prudence by its very nature is more opposed to sin which arises from a disorder of the appetite reply to objection three sinners can take good counsel for an evil end or for some particular good but they do not perfectly take good counsel for the end of their whole life since they do not carry that counsel into effect hence they lack prudence which is directed to the good only and yet in them according to the philosopher ethics 612 there is cleverness that is natural diligence which may be directed to both good and evil or cunning which is directed only to evil and which we have stated above to be false prudence or prudence of the flesh 14th article whether prudence is in all who have grace objection one you would seem that prudence is not in all who have grace prudence requires diligence that one may foresee a right what has to be done but many who have grace have not this diligence therefore not all who have grace have prudence objection 2 further a prudent man is one who takes good counsel as stated above in article 8 objection 2 and in article 13 objection 3 yet many have grace who do not take good counsel and need to be guided by the counsel of others therefore not all who have grace have prudence objection 3 further the philosopher states in the topics 3 2 that young people are not obviously prudent yet many young people have grace therefore prudence is not to be found in all who have grace on the contrary no man has grace unless he be virtuous now no man can be virtuous without prudence for Gregory says on the morals of Job 246 that the other virtues cannot be virtues at all unless they affect prudently what they desire to accomplish therefore all who have grace have prudence I answer that the virtues must needs be connected together so that whoever has won has all as stated above parts prima secunde question 65 article 1 now whoever has grace has charity so that he must needs have all the other virtues and hence since prudence is a virtue as shown above in article 4 he must of necessity have prudence also reply to objection 1 diligence is twofold one is merely sufficient with regard to things necessary for salvation and such diligence is given to all who have grace whom is unctioned teacher of all things 1st John 227 there is also another diligence which is more than sufficient whereby a man is able to make provision both for himself and for others not only in matters necessary for salvation but also in all things relating to human life and such diligence as this is not in all who have grace reply to objection 2 those who require to be guided by the counsel of others are able if they have grace to take counsel for themselves in this point at least that they require the counsel of others and can discern good from evil counsel reply to objection 3 acquired prudence is caused by the exercise of acts where for its acquisition demands experience and time ethics to one hence it cannot be in the young neither in habit nor an act on the other hand gratuitous prudence is caused by divine infusion where for in children who have been baptized but have not come to the use of reason there is prudence as to habit but not as to act even as an idiot whereas in those who have come to the use of reason it is also as to act with regard to things necessary for salvation this by practice merits increase until it becomes perfect even as the other virtues hence the apostle says in Hebrews 514 that strong meat is for the perfect for them who by custom have their senses exercised to the discerning of good and evil 15th article whether prudence is in us by nature objection 1 you would seem that prudence is in us by nature the philosopher says that things connected with prudence seem to be natural namely cinesis gnomes note cinesis and gnomes confer pars prima secunde question 57 article 6 and the like but not those which are connected with speculative wisdom now things belonging to the same genus have the same kind of origin therefore prudence also is in us from nature objection 2 further the changes of age are according to nature now prudence results from age according to Job 1212 in the ancient is wisdom and in length of days prudence therefore prudence is natural objection 3 further prudence is more consistent with human nature than with that of dumb animals now there are instances of a certain natural prudence in dumb animals according to the philosopher on his histories of animals 81 therefore prudence is natural on the contrary the philosopher says in ethics 21 that intellectual virtue is both originated and fostered by teaching it therefore demands experience and time now prudence is an intellectual virtue as stated above in article 4 therefore prudence is in us not by nature but by teaching and experience I answer that as shown above in article 3 prudence includes knowledge both of universals and of the singular matters of action to which prudence applies the universal principles accordingly as regards the knowledge of universals the same is to be said of prudence as of speculative science because the primary universal principles of either are known naturally as shown above in article 6 except that the common principles of prudence are more conatural to man for as the philosopher remarks in ethics 10 7 the life which is according to the speculative reason is better than that which is according to man whereas the secondary universal principles whether of the speculative or of the practical reason are not inherited from nature but are acquired by discovery through experience or through teaching on the other hand as regards the knowledge of particulars which are the matter of action we must make a further distinction because this matter of action is either an end or the means to an end now the right ends of human life are fixed where for there can be a natural inclination in respect of these ends thus it has been stated above in paris prima secunde question 51 article 1 and question 63 article 1 that some from a natural inclination have certain virtues whereby they are inclined to write ends and consequently they also have naturally a right judgment about such like ends but the means to the end in human concerns far from being fixed are of manifold variety according to the variety of persons and affairs where for since the inclination of nature is ever to something fixed the knowledge of those means cannot be in man naturally although by reason of his natural disposition one man has a greater aptitude than another in discerning them just as it happens with regard to the conclusions of speculative sciences since then prudence is not about the ends but about the means as stated above article 6 and also paris prima secunde question 57 article 5 it follows that prudence is not from nature reply to objection one the philosopher is speaking there of things relating to prudence insofar as they are directed to ends where for he had said before in ethics 6511 that they are the principles of the eugenica note literally for the sake of which are the means namely the end and so he does not mention oibulia among them because it takes council about the means reply to objection to prudence is rather in the old not only because their natural disposition columns the movement of the sensitive passions but also because of their long experience reply to objection three even in dumb animals there are fixed ways of obtaining an end where for we observe that all the animals of a same species act in like manner but this is impossible in man on account of his reason which takes cognizance of universals and consequently extends to an infinity of singulars 16th article whether prudence can be lost through forgetfulness objection one you would seem that prudence can be lost through forgetfulness for since science is about necessary things it is more certain than prudence which is about contingent matters of action but science is lost by forgetfulness much more therefore is prudence objection to further as the philosopher says in ethics 23 the same things but by a contrary process in gender and corrupt virtue now the engendering of prudence requires experience which is made up of many memories as he states at the beginning of his metaphysics one one therefore since forgetfulness is contrary to memory it seems that prudence can be lost through forgetfulness objection three further there is no prudence without knowledge of universals but knowledge of universals can be lost through forgetfulness therefore prudence can also on the contrary the philosopher says in ethics 65 that forgetfulness is possible to art but not to prudence i answer that forgetfulness regards knowledge only where for one can forget art and science so as to lose them all together because they belong to the reason but prudence consists not in knowledge alone but also in an act of the appetite because as stated above in article 8 its principal act is one of command whereby a man applies the knowledge he has to the purpose of appetite and operation hence prudence is not taken away directly by forgetfulness but rather is corrupted by the passions for the philosopher says in ethics 65 that pleasure and sorrow pervert the estimate of prudence where for it is written in daniel 1356 beauty hath deceived thee and lust hath subverted thy heart and in exodus 238 neither shall thou take bribes which blind even the prudent note the duty translation has prudent stated as wise nevertheless forgetfulness may hinder prudence insofar as the latter's command depends on knowledge which may be forgotten reply to objection one science is in the reason only hence the comparison fails as stated above confer bars prima secunde question 53 article 1 reply to objection 2 the experience required by prudence results not from memory alone but also from the practice of commanding a right reply to objection 3 prudence consists chiefly not in the knowledge of universals but in applying them to action as stated above in article 3 where for forgetting the knowledge of universals does not destroy the principal part of prudence but hinders it somewhat as stated above end of question 47 part 3 read by michael shane craig lambert lc question 48 a summa theologica secunde secunde triates on the cardinal virtues the virtue of prudence this is a liberox recording all liberox recordings are in the public domain for more information or to volunteer please visit liberox.org summa theologica secunde secunde triates on the cardinal virtues the virtue of prudence by saint thomas equinas translated by the fathers of the english dominican province question 48 of the parts of prudence in one article we must now consider the parts of prudence under which head there are four points of inquiry one which are the parts of prudence two of its integral parts three of its subjective parts four of its potential parts article whether three parts of prudence are fittingly assigned objection one he would seem that the parts of prudence are assigned unfittingly tolly in his rhetoric's 253 assigns three parts of prudence namely memory understanding and foresight macrobius following the opinion of plotinus ascribes to prudence six parts namely reasoning understanding circumspection foresight docility and caution Aristotle says in ethics 6 9 10 and 11 that good counsel sinesis and genome belong to prudence again under the head of prudence he mentions conjecture shrewdness sense and understanding and another greek philosopher note andronicus confer question 80 objection 4 says that 10 things are connected with prudence namely good counsel shrewdness foresight reignitive note reignitiva military political and domestic prudence dialectics rhetoric and physics therefore it seems that one or the other enumeration is either excessive or deficient objection 2 further prudence is specifically distinct from science but politics economics logic rhetoric physics are sciences therefore they are not parts of prudence objection 3 further the parts do not exceed the whole now the intellect of memory or intelligence reason sense and facility belong not only to prudence but also to all the cognitive habits therefore they should not be set down as parts of prudence objection 4 further just as counseling judging and commanding are acts of the practical reason so also is using as stated above in par's prima secundi question 16 article 1 therefore just as oybulia which refers to counsel is connected with prudence and scenicis and genome which refer to judgment so also at something to have been assigned corresponding to use objection 5 further solicitude pertains to prudence as stated above in question 47 article 9 therefore solicitude also should have been mentioned among the parts of prudence I answer that parts are of three kinds namely integral as wall roof and foundations are parts of a house subjective as ox and lion are parts of animal and potential as the nutritive and sensitive powers are parts of the soul accordingly parts can be assigned to a virtue in three ways first in likeness to integral parts so that the things which need to concur for the perfect act of a virtue are called the parts of that virtue in this way out of all the things mentioned above eight may be taken as parts of prudence namely the six assigned by macrobius with the addition of a seventh notably memory mentioned by tully and eustochia or shrewdness mentioned by Aristotle for the sense of prudence is also called understanding where for the philosopher says in ethics 611 of such things one needs to have the sense and this is understanding of these eight five belong to prudence as a cognitive virtue namely memory reasoning understanding docility and shrewdness while the three others belong there too as commanding and applying knowledge to action namely foresight circumspection and caution the reason of their difference is seen from the fact that three things may be observed in reference to knowledge in the first place knowledge itself which if it be of the past is called memory if of the present whether contingent or necessary is called understanding or intelligence secondly the acquiring of knowledge which is caused either by teaching to which pertains docility or by discovery and this belongs to eustochia that is a happy conjecture of which shrewdness is a part which is quick conjecture of the middle term as stated in posterior analytics one nine thirdly the use of knowledge in as much as we proceed from things known to knowledge or judgment of other things and this belongs to reasoning and the reason in order to command a right requires to have three conditions first to order that which is befitting the end and this belongs to foresight secondly to attend to the circumstances of the matter in hand and this belongs to circumspection thirdly to avoid obstacles and this belongs to caution the subjective parts of a virtue are its various species in this way the parts of prudence if we take them properly are the prudence whereby a man rules himself and the prudence whereby a man governs a multitude which differs specifically as stated above question 47 article 11 again the prudence whereby a multitude is governed is divided into various species according to the various kinds of multitude there is the multitude which is united together for some particular purpose thus an army is gathered together to fight and the prudence that governs this is called military there is also the multitude that is united together for the whole of life such is the multitude of a home or family and this is ruled by domestic prudence and such again is the multitude of a city or kingdom the ruling principle of which is cognitive prudence in the ruler and political prudence simply so called in the subjects if however prudence be taken in a wide sense as including also speculative knowledge as stated above in question 47 article 2 second reply then its parts include dialectics rhetoric and physics according to three methods of prudence in the sciences the first of these is the attaining of science by demonstration which belongs to physics if physics be understood to comprise all demonstrative sciences the second method is to arrive at an opinion through probable premises and this belongs to dialectics the third method is to employ conjectures in order to induce a certain suspicion or to persuade somewhat and this belongs to rhetoric it may be said however that these three belong also to prudence properly so called since it argues sometimes from necessary premises sometimes from probabilities and sometimes from conjectures the potential parts of a virtue are the virtues connected with it which are directed to certain secondary acts or matters not having as it were the whole power of the principle of virtue in this way the parts of prudence are good counsel which concerns counsel cinesis which concerns judgment in matters of ordinary occurrence and genome which concerns judgment in matters of exception to the law while prudence is about the chief act notably that of commanding reply to objection one the various enumerations differ either because different kinds of parts are assigned or because that which is mentioned in one enumeration includes several mentioned in another enumeration thus tully includes caution and circumspection under foresight and reasoning the cility and shrewdness under understanding reply to objection two here domestic and civic prudence are not to be taken as sciences but as kinds of prudence as to the other three the reply may be gathered from what has been said reply to objection three all these things are reckoned parts of prudence not by taking them all together but in so far as they are connected with things pertaining to prudence reply to objection four right command and right use always go together because the reasons command is followed by obedience on the part of the lower powers which pertain to use reply to objection five solicitude is included under foresight end of question 48 read by michael shane craig lambert lc


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Hello, KubeCon. My name is Wojtek Teczyński. I work in Google. And I'm part of this amazing Kubernetes community for almost six years at this point. Amongst others, I'm the TL of six-scale ability. And in connection to that, during this presentation, I'm going to talk about example features that allow us to run Kubernetes clusters with 15,000 nodes. But let's start with making this explicit. Kubernetes clusters with 15,000 nodes are already a thing. As you might have heard, earlier this year, Biocrop Science and Google together published an interesting blog post. You can read there how, thanks to running on GKE clusters with 15,000 nodes, with almost a quarter million of cores and over 1.5 petabytes of RAM. Biocrop Science was able to process roughly 15 billion genotypes per hour as part of the pipeline responsible for deciding which seeds, which are the final product, should be advanced for further experiments in their R&D department. Couple months later, during joint presentation, Twitter and Google together described how we approached validating that their applications that they are currently running on their Mesas Aurora clusters on prem can be split and run on GKE clusters with 15,000 nodes. But the goal of this presentation is to show that this work matters for all of you, even if your clusters are order, or maybe even orders of magnitude smaller. However, we need to start with understanding what scalability really means for Kubernetes. Even though we tend to use node count or cluster size as a proxy for the overall scale, it's actually much more complex than that. Scalability is a multi-dimensional problem with dozens of dimensions, such as number of services, number of volumes, pod churn, and so on. Even though, in many use cases, they may scale together with cluster size, you need to acknowledge that node count is not the only thing that matters. So how did we approach scaling Kubernetes to the next level, which 15,000 nodes certainly is? The core principle for any scalability or performance work is don't optimize blindly and always solve real-life problems. Almost every optimization is making a system a little bit more complex, so it's super important to keep that in mind. According to this rule, we first found users for whom running such a massive clusters would have actual benefits. We started with understanding their use cases, but also their motivations. And as you may suspect, those are the ones that I already mentioned. The first one was buyer crop science, who is using Kubernetes to run embarrassingly parallelizable batch computations. For them, larger clusters immediately translate to having results faster. In addition to that, given that the users are generally data scientists who don't want to understand the underlying infrastructure, they want to make it as simple as possible, ideally a single cluster. The second one was Twitter, who would like to migrate the microservices apps to Kubernetes. As I already mentioned, they are currently running them on Mesas Aurora clusters, which can handle even 40 to 60,000 nodes in a single cluster. So while migrating to Kubernetes, they would like to avoid the need to suddenly manage an order of magnitude more clusters. In addition to that, they would also like to unify their setup and run applications that they are currently running differently, like some stateful apps also on Kubernetes. At this point, you are probably thinking that you are wasting your time. I'm saying that we focused on two users. Doesn't that sound simple, by the way? And your workloads probably aren't even similar to what they are running. However, just like landing on the moon will push the frontiers of technology, it's the same with our scalability work. The improvements we did not only push the scalability limits, but they are primarily making all clusters more reliable and more performant. So now I'm going to describe a couple improvements to show how they make your life better. So let's start with HCD. The main HCD improvement that is worth mentioning is concurrent reads. Before this change, every read operation on HCD was blocking all other operations, both read and write, for the entire time of its processing. Thanks to this improvement, we are now blocking other operations only for a very short period of time just to grab a copy on write pointer to the current state. While it's crucial for large clusters, it's not the only place where it helps. Imagine that you have 10,000 custom resources of your CRD, and I've seen real production clusters like that with just 10 or 20 nodes. Thanks to this improvement, when you are listing custom resources, you will no longer observe spikes in the API co-latencies even if your cluster is small. And it's already available in HCD 3.4. So let's now make a step up and talk about API server and API machinery. As you know, watch is the crucial part of our API. But no request runs forever. So what happens when it times out, especially if it's a very selective watch that selects only a small fraction of all objects of a given type? To resume the watch, we are using resource version of the last object that we received via it. But many other changes might have happened in the meantime. And now we need to process all of them again, simply because we didn't have a way to signal the client that they have already been processed. So we introduced a concept of watch bookmark. It's basically a new event type that you can receive via watch that basically tells you we processed everything up to a resource version X. So if you didn't receive anything, it simply means nothing much to your selector. As you can see, there is nothing specific to 15,000 node clusters. It helps for clusters with 1,000 or even 100 nodes. And it just happens out of the box because Kubelet that is watching its own pods is a perfect example of very selective watch that benefits a lot from this improvement. And it's already GA in Kubernetes 117. However, improvements were needed across the whole stack. So let's take a look into an example from the networking area. For those not familiar with Endpoints API, the Endpoints object contains information about all backends of a given service. So its size is proportional to the number of pods behind that service. That has many consequences, but let's think about Kube Proxy, which is an agent running on every single node in the cluster that is responsible for programming in cluster networking. In order to do that, it's watching for changes of every single Endpoints object in the cluster. So imagine that you have a service with 5,000 pods. The corresponding Endpoints object will have, optimistically, around 1 megabyte. So in 5,000 node cluster, and I'm not even talking about 15,000 nodes at this point, that means that API server has to send 5 gigabytes of data for a single change of that object. And for the rolling upgrade of that service, it would be 25 terabytes of data. To mitigate that, we introduced the concept of Endpoint Slice that allows us to share the information about the Endpoints of a given service into multiple Endpoint Slice objects. Thanks to that, we can significantly reduce the load on the control plane and amount of data that API server has to serialize and send. The pieces of that solution went better in Kubernetes 119. So let's take a look into one more example, this time from the storage area. And let's talk about secrets and config maps. As you probably know, whenever any of those changes, cubelets are reflecting those changes to all pods that are mounting them. But in order to do that, they are opening a watch for every single secret and config map that the pods are using. That may translate even to hundreds or hundreds of thousands of additional watches in the system. Optimizing them would bring a lot of complexity to the API machinery. And as I mentioned before, we should always be solving real life problems. So we started talking to users, and it appeared that they don't mutate majority of secrets and config maps at all. So we introduced the concept of immutability to secret and config map API. When explicitly marked as immutable by the user, their contents cannot be changed. But cubelets also don't need to watch for them, which is vastly reducing the load on the control plane. As in the previous examples, there is nothing specific to large clusters. And you can take advantage of it even if your cluster is small. And it's already beta in Kubernetes 119. However, we were also looking outside of core Kubernetes. We worked closely with Golang community on optimizing its memory locator. It may be surprising to many of you, but log contention at the level of Go memory allocator is actually one of the bottlenecks that we really suffer from. While some optimizations have already landed in newer versions of Go, even more are coming. And this benefits not just Kubernetes. It benefits everyone who is writing their applications in Go. I described a couple improvements, and all of them, as well as tens or maybe even hundreds of others, were done in upstream Kubernetes. However, that doesn't immediately mean that every Kubernetes distribution will scale to 15,000 node clusters. In order to work, your ecosystem has to work at that scale too. That includes the underlying infrastructure, both compute and networking. You need 15,000 VMs or machines and networking between them. It includes auto scaling, logging and monitoring, control plane upgrades, and many other things. Based on GKE experience, I can say that it's a huge effort to make all of them work. So Kubernetes improvements are necessary, but they don't solve all the problems for you. There is one more question that we should answer here, which is how do we know when we can stop? Fortunately, the answer for this question is fairly simple. As soon as we meet our SLOs, Service Level Objectives, you can think about them as system level metrics with thresholds. While the concepts they cover in Kubernetes are still fairly basic, like API co-latencies or pod startup type, they greatly correlate with user experience. So to summarize, scalability work matters for almost everyone, because scalability is much more than just the cluster size. The improvements we did to push the scalability limits are also, or maybe even primarily, making smaller clusters more reliable and more performant. So if your cluster didn't work because of some scalability or performance issues in the past with Kubernetes 118 and higher versions, it's probably time to reevaluate it. Unfortunately, we don't have more time, so I will just mention that with upcoming releases, you can expect even more improvements and extending the portfolio of use cases supported by clusters with 15,000 nodes. And with that, thank you very much for staying with me.