So my fiance Sharon Berry posted a very early (like 2 years early) draft of her thesis on a wiki here. The broad question she is addressing is how we can come to have accurate mathematical knowledge which I figured might be of interest to some people who check out the philosophy part of my blog. I also figured I’d take this chance to share my own thoughts on the subject. However, to give credit where credit is due I would never have really thought through these issues if Sharon hadn’t brought up the subject and many of the ideas are really hers. However, I take them in a very different direction than she does.

The really short version of my attitude to the problem of mathematical knowledge is “What Problem?” I mean obviously mathematical knowledge is subject to the same skeptical doubts that other forms of knowledge are but I’m unconvinced that there is any particular problem unique to mathematical knowledge. More specifically I would say that mathematical knowledge is nothing but a limiting case of other sorts of knowledge so it poses no problem over and above the problem of understanding the meaning and our knowledge of other sorts of statements. Of course explaining meaning is a notoriously difficult problem in it’s own right but I’m tempted to think that it’s a hopeless problem. Ultimately one must merely take meaning to be a primitive concept but that’s another discussion.

I need to get back to working on my thesis so I won’t give more than a very very quick sketch of my thoughts here but roughly I take it there are two primary reasons one might think that mathematical knowledge requires special explanation.

1. The Benacerraf problem: How could we come to know anything about numbers if they don’t have causal powers, we don’t interact with them and so forth.
2. How could it be that our mathematical theories turn out to be useful in the way they are.

Platonism and Reference

So if one accepts a platonic theory of mathematical meaning then there may indeed be special problems about mathematical knowledge. That is if the meaning of a statement like 2+2 =4 is really that some special 2 object out there bears a certain relation to itself and the four object one might wonder how it is that we come to know about these platonic objects. However, I’m inclined to simply turn the question around and ask whether the platonic theory in question provides any reason to think that “2″ refers to something we would ‘recognize’ as an integer or whether it could (logically not metaphysically) be that 2 refers to the concept of bunny rabbits and all our statements about arithmetic are really nonsensical. If the platonic interpretation of mathematics tells us that the reference of two must really behave like 2 to qualify as the correct reference then we know exactly how we come to have true beliefs about the numbers — because if our beliefs weren’t largely true we would be talking about something else[^enough]. On the other hand if we don’t have any restrictions about what sort of platonic object 2 might refer to then we aren’t justified in adopting this kind of theory in the first place.

Unfortunately the debate about Platonism and competing philosophies of mathematical largely distracts from what I think are the important issues. As I’ve argued previously Platonism in and of itself says very little about mathematics. What the last paragraph as well as my previous post on the issue emphasize is that it isn’t really Platonism that is doing the work it is your theory of reference. Really on it’s own Platonism says nothing very significant¹, it’s the means by which our talk maps to particular platonic objects that really does the work in the theory. This raises the obvious question of what we even mean when we say that the reference of a certain term is such and such. Are we merely making a claim about dispositions and talk or are we invoking some real metaphysical relation. While Platonism provides a good motivation to consider the issue I think a proper examination of this question of what sort of thing the meaning relation is in the first place illustrates the non-problem of philosophy of mathematics in general.

Platonic Realism About Reference

There are two ways one could understand claims about meaning and reference One could think that the relation of meaning is a truly objective notion with metaphysical substance. That is that the relation between words/mental states/speaking contexts is some and references/meanings is something like a platonic entity in it’s own right. On such a theory it is presumably logically possible (but not metaphysically) that when I say “2″ it really (by virtue of this objectively existing meaning relation) refers to rabbit. In other words the meaning of word is a notion much like the moral status of an action under on a realist moral theory.

Just as with moral realism I think the appropriate response to this notion of meaning is to challenge that it counts as meaning at all. Ultimately there is just this relation out there mapping situations/worlds/utterances/mental states/whatever to references/intentions but why should we think this picks out what we talk about when we use the term meaning? Additionally on this sort of platonic realism about meaning we don’t have any reason to actually believe that we really do have knowledge. After all maybe the objective meaning relation isn’t what we think it is at all and what we take to be true statements aren’t true at all.

One might still be tempted to insist that obviously we have knowledge thus the fact that this theory can’t explain this fact is a puzzle requiring explanation. However, this simply gets things backwards and implicitly rejects the very assumptions of the theory itself. If we accept this sort of theory we need to just bite the bullet and say we don’t know if we really know anything and thus how we know things doesn’t require explanation. Personally I think our intuition that our usage determines meaning is a good reason to reject this sort of theory but in either case this leaves no special problems for mathematics. Of course you might try and say that the mapping between statements and meanings/references must obey certain restrctions but this does no good at all since of course any actual map will have some facts that are true of it but this does nothing to offer us reason to think we have any knowledge of what they are.

Naturalist Theories of Meaning

I think a much more promising approach² is to jettison all the metaphysical baggage and start from the assumption that meanings, ultimately must be defined in terms of sounds, dispositions, actions and other arrangements of matter. That is nothing special or magical goes on with meanings. They are just a concept introduced to organize very complicated descriptions of human behavior in terms of atoms and physical laws. Thus the ultimate standard against which we judge a theory of meaning is it’s predictive accuracy and theoretical utility (how well does it work with other models we wish to use). In some sense already this approach should suggest that there shouldn’t be any deep paradoxes in terms of meaning. After all we are confident that the description of human behavior at the level of atoms is consistant thus any apparent difficulty at the level of meanings either reflects a confusion on our part or a poor choice of definition.

To put the point a bit differently we should think about a theory of meaning much the way we think about thermodynamics as derived from statistical mechanics. Yes, it can be a powerful theory with useful concepts and important impacts but ultimately just as debates about whether entropy is the log of the number of possible states holding X,Y and Z fixed or just X and Y doesn’t reflect any fundamental fact about the universe but a definitional choice we make that is judged on it’s utility. Thus theories like fictionalism or formalism shouldn’t be understood as making different philosophical claims but rather judged simply on their utility in predicting how people actually use words. Indeed one might very well conclude that different models are most appropriate in different circumstances.

Ultimately then the question about how we can come to have mathematical knowledge is largely a non-question. I can point to the actual ways that mathematicians prove theorems and reach conclusions and that right there shows how we come to have mathematical knowledge. Still one might ask but why are the results of our proofs actually true? However, this has a totally trivial answer. The reason that proofs give us true mathematical results is that every step of the proof is truth preserving. Indeed we can go through this and using the fact (in the meta-language) that A and B is true if and only if A is true or B is true show that the methods mathematicans use to reach theorems really do produce true theorems. Asking for anything more is a demand to know why logic is true. Obviously at a very basic level we have to just assume that logic is true (see Quine’s arguments about this point in his discussions of radial translation) so it’s unclear what is left to be explained at all.

To put the point slightly differently it’s contradictory to worry about how we get mathematics correct. Either the question tells us how we have reason to believe we do get mathematics right, in which case it tells us the answer or it offers no such explanation and we have no need to explain a phenomena that we don’t have reason to accept as true.

Usefullness of Mathematics

This finally brings us to the question of why mathematics turns out to be useful. One might think that it’s surprising that the results of mathematics tells us useful things about the world. Certainly in one sense it is surprising, but that’s the sense in which the understandability of the world is surprsing, i.e., that induction works. While it may appear that mathematics directly makes predictions about the world (if I have two apples in my bag and place another two apples into my bag I have four apples in my bag) in fact it’s only the combination of mathematical theorems with contingent bridge laws that makes these predictions (apples don’t appear or disappear when I place more of them together). One might try and minimize the significance of these bridge laws by saying something like “So long as apples don’t appear or disappear the number of apples in my bag is the number of apples I added minus the number I removed.” However, this merely begs the question by working in our expectation that the plus operation on the natural numbers describes how objects behave into the definition of appear or disappear. I could equally well claim that apples were disappearing and reappearing all the time but if they didn’t do so we could see that adding n apples to a bag with m apples in it results in a bag with n x m apples in it.

In fact the usefulness of something like mathematics is an easy consequence of a well known theorem in recursion theory. Supposing we have a language complex enough to express arbitrary procedures³ then that language will contain infinitely many different ways to state the same procedure, some subset of which will be possible to construct a verification that they are equivalent. In other words no matter how weird your language is you can’t get around the fact that some things will turn out to be non-obviously equivalent which suggests that it will be useful to have a means to identify at least some of them.

Usefullness and Knowledge

The final worry is that one might try and link the two concepts and ask how it is that we come to have mathematical knowledge that yields useful results. Thus even if we don’t have abstract reasons to believe that the syntactic manipulations of mathematicians meet some independent standard of being true we do notice that they let us build rockets and cure disease and the like. Thus one might think the utility of mathematics requires some explanation.

Once again though I think a careful examination of the question reveals it to be a non-worry. If by mathematics you merely mean the sort of thing that mathematicians do then it’s undeniable that what counts as mathematics is partially determined by what is useful. While many types of mathematics are very abstract the subject in the large is influenced by what has solved problems presented by the world. This point is made even more forcefully if you try to define mathematics as any abstract rule based manipulation of symbols. After all under such a definition certain types of astrology would qualify which most assuredly is not useful. Similarly any other means by which you tried to formally define the problem is likely to either reduce to triviality or not call our for any explanation at all.

This was a pretty hurried and scattered explanation of my thouhts so hopefully people ould follow it. If you are confused but curious about what I’m trying to say anyway feel free to post a comment or ask me via email

UPDATE: Fixed some wording, added clarification at the bottom.

Consider a philosophical debate between two materialists over whether a virus is alive. Philosopher A advances the hypothesis that any organism capable of manipulating it’s environment to copy itself is alive. Philosopher B counters that Shakespeare’s “Macbeth” has this property as it’s induces people to produce reproductions of the work and instead argues that a being is alive only if it doesn’t require outside intervention to duplicate itself. Philosopher A counters with an example of a plant that humans have cultivated for long enough that it is now incapable of reproduction without intentional human intervention.

Now we can imagine this debate continuing as both philosophers refine their theories as to what constitutes life but no matter how long they argue in this fashion they can’t hope to reach any substantive conclusions. Why? Because they didn’t disagree about anything but word usage at the outset. As materialists they both reject the notion of some elan vital that we might add to our fundamental ontology to ‘explain’ what counts as alive and what doesn’t. As far as the virus goes they would both accept the biologists explanation as to how it reproduced they only disagree on how to label this event. Now there are certainly times it makes sense to argue about labeling but we naturally expect such debates to take a very different form. In particular when people merely disagree about how we should term something they will usually sidestep the debate eventually and simply qualify their wording. When people disagree on how in fact people are inclined to use words they tend to either shrug and move on or to start pulling out real empirical data¹. In no case where people understand themselves to be merely debating a matter of labeling would we expect them to argue about the issue for decades in reputable journals with no hint that they view themselves as debating some empirical fact about usage or pragmatic fact about what would make for good usage.

Unfortunately there seems to be no shortage of arguments in philosophy that can’t be explained as anything other than a confusion of a question of terminology as a substantive question². There are a host of examples but let me give a couple

Is formalism or fictionalism the right philosophy of math?

Now there is (arguably) a genuine substantive question as to whether mathematical Platonism is true. Is there or is there not a realm of platonic objects out there? And if you believe in a substantive notion of reference (reference facts aren’t ontologically reducible to physical/mental facts) whether or not that is what we refer to with mathematical talk. However, when we get down to debates between functionalism and fictionalism things suddenly become much more unclear.

Neither theory disagrees about what in fact mathematicians assert nor makes any fundamentally different metaphysical suppositions. Nor do the two theories compete on genuine empirical predictions. Neither theory attempts to best predict what in practice people will tend to assert about mathematics. So in what sense can there be said to be a substantive question at issue here? And if not why believe these two interpretations are in opposition to each other?.

The “Proper” Conception of Evidential Support

In formal epistemology there seems to be a great deal of ink spilt arguing over what the ‘proper’ notion of confirmation is. Now Brandon Fitelson has always (at least to my eyes) pushed for a ‘there is no fact of the matter’ resolution to many debates in this area so I wouldn’t accuse him of making this sort of mistake but this paper of his gives a nice picture of what sorts of arguments are at play in the area. In particular there seems to be a long lasting dispute as to what the ‘right’ notion of confirmation turns out to be. Is it a three place relation between evidence, theory 1 and theory 2 or is it a two place relationship between evidence and a hypothesis?

Now I would be most surprised if anyone in this debate thought confirmation was an substantive notion (but I’ve been wrong about this sort of thing before), that is that when we assert that evidence E favors hypothesis H we aren’t just asserting some fact about probabilities, models or events but actually claiming that there is some special ‘confirmation’ property in our ontology that adheres to that particular relation but not to others that we might have chosen instead to term confirmation and that. Yet if we aren’t being ontologically liberal like this it would seem that all this debate about what is the ‘right’ notion of confirmation seems silly. We can all agree on the formal consequences of each notion and just set aside the contentious terminological question³.

Is Welfare Desire Satsifaction or Utility Maximization

I mention this because it was the argument that first made me realize that many of these debates couldn’t be substantive. While many people want to add an extra ontological fact to explain morality few people are inclined to indiscriminately add ontological entities for subsidiary moral concepts like welfare yet they are perfectly happy to debate the issue as if it were substantive.

In particular many people argue about whether we maximize welfare by maximizing utility or by maximizing desire satisfaction (or something else) as if it was a separate question we resolve prior to figuring out what is morally good. However, short of proposing a new fundamental property or relation it would seem that the debate over what increases someone’s welfare is merely a terminological question.

Kripke’s Causal Theory of Reference for naturalists

I debated about including this one since some people who buy into Kripke’s theory believe it as a genuine substantial claim. That is they add extra fundamental objects to their ontology (references, meanings etc..) and make the substantial claim that somehow our intuitive explanations of words in terms of other words track these objects and that as a real ontological fact it turns out that the reference of our word is determined by it’s causal history.

However, many of the people who take these theories seriously would call themselves naturalists or physicalists and would balk at the suggestion that by endorsing Kripke’s theory they were making grandiose metaphysical claims that couldn’t possibly be explained in terms of anything physics could in principle ever discover. Presumably as a physicalist one should accept the fact that there is nothing more to speech acts than certain configurations of matter and that there is no free floating metaphysical object ‘the reference’ that exists over and above the configuration of matter. As I’ve argued before it’s absurd for a good naturalist/physicalist to have a horse in the internalist/externalist debate except insofar as one turns out to be a genuinely better empirical predictor of future events.

My thesis is that there is a strong bias towards taking mere disputes about terminology and assuming that they are substantive. Not only is it a tempting fallacy to fall into on it’s own but it also creates for a much more interesting seeming discussion. It seems much more significant to say that one is figuring out the nature of life than to admit one is merely debating what we should call “life.” In any case whatever the reason it seems that this is a common fallacy that I see in philosophical discourse and one we should guard against. There are more than enough substantive arguments to keep philosophers busy and some of these non-substantive arguments are worth having as well but which arguments we take to be persuasive will be very different once we understand it is merely a terminological debate. Importantly once we accept that many of our debates are merely terminological we can no longer assume that there is any tension between things like internalism and externalism or different measures of confirmation.

As an aside I think this is in some sense the issue between Carnap and Quine over the nature of analyticity but that’s something for another post.

CLARIFICATION: I don’t want to claim that these debates couldn’t be rendered substantive. Really all I want to claim is that they are not naively substantive questions so using a standard of argumentation suited to this assumption is in error. I don’t mean to say that we need to abandon these questions only that we should figure out what we are trying to say and what it would take to establish our claim before we try to argue for one side or another.

Also I’m not convinced that any particular philosopher is making this error. It often seems that when I talk to any given philosopher about the matter they have some complex alternative interpretation of the claims at issue that either recognizes them as not substantive or renders them so through some non-obvious interpretation. It’s entirely possible that this is merely a process error but at the very least something is wrong when people adopt the form of a substantive argument for notions that don’t seem like they could be substantive without giving an indication as to what way it is (non-obviously) substantive (least different people think they are debating different questions). What I really want to do hear is not so much to advance my particular theory as to what is going wrong but to call attention to the fact that something seems really out of wack (or have someone give me a satisfactory explanation as to why it is not).

So today on philosophy talk they invited Alan Wood to talk about Hegel. As usual I was impressed by Ken Taylor and John Perry (the hosts) who tried valiantly to work out the content of Hegel’s claims. For instance they tried to pin Professor Wood down on what the rationality of history really meant and if it was just a complicated way to say that the people who write history will always impose a narrative structure (answer was no). However, as much as I like the hosts the attitude they took toward Hegel is a perfect example of the problem with philosophy today. This quote from the program description is a perfect example.

Georg Wilhelm Friedrich Hegel is without doubt one of the most influential philosophers of all time. He has, however, been largely ignored by American “analytic” philosophers of the twentieth century. John in particular, and Ken to a lesser extent, don’t know nearly as much about Hegel and his philosophy as they should.

Note the clear assumption in the description that Hegel is worth learning something about. Sure as an influential philosopher it may be worthwhile to read the cliff notes for cultural literacy and have enough familiarity with his work to decide if he is worth further investigation but as the comments made during the program indicated this assumption of merit goes far beyond that. The hosts of philosophy talk are just doing the same thing I see almost all philosophers do at conferences, assuming that any philosophy that is or was academically studied is worthwhile. The defacto attitude in philosophical circles seems to be that anything recognized as genuine philosophy (not a layman’s ‘philosophy’ of life) has continuing value and if you don’t see the merit in their ideas it is because you don’t understand them.

The fact that philosophers still put great value on reading the original contributors has always made me suspicious that social (and perhaps literary) considerations were interfering with the quest for the truth (more on this later) but the example of Hegel puts the matter beyond any doubt. Just the sheer number of incompatible interpretations of Hegel should already make on suspicious of Hegel’s worth. It seems totally plausible that the lack of clear content in Hegel is letting people read their own ideas into his writings. If I was to publish a blank book I should clearly not be given credit for whatever people happen to write in it and similarly we should credit Hegel’s works (not the historical person) as saying only that the truth lies somewhere in the range of plausible interpretations. In other words if the cost in effort of extracting good arguments and theories from Hegel is the same as the effort to come up with those on your own then Hegel is no more valuable than a blank book.

The real damning piece of evidence though is that we know Hegel’s methods and philosophical ‘analysis’ produces complete nonsense. Several good examples are given on this page. Among other things Hegel claimed to have proved (false) facts about the location of the planets from pure philosophical considerations. Popper has pointed out that many of Hegel’s statements appear to be little more than sheer gibberish or when intelligeable completely trivial facts dressed up in fancy words. Thus our situation with Hegel is much the same as it would be if we were talking to a crazy man on the street. If we read parts of Hegel and find they don’t make sense to us or the arguments and claims seem unconvincing we shouldn’t give him the benefit of the doubt and assume it is our failing for not understanding him. We know that Hegel often writes complete gibberish and claims to have shown obviously false facts so we have good reason to suspect other passages of his suffer from similar problems.

In order to progress philosophy, like other disciplines trying to uncover the truth, needs to effectively winnow out invalid and unproductive theories. Since philosophy has less external checks on its theories than mathematics or science if anything philosophers need to be even more vigilant to make sure that inertia or social pressure doesn’t allow invalid or nonsensical theories to persist. Unfortunately as this example with Hegel illustrates the discipline is unfortunately doing just the opposite. By making the assumption that anything some philosophers study has important worthwhile things to say even if all the claims we can evaluate are false or non-sense we virtually guarantee that bad philosophy is never winnowed away. Since anyone who hasn’t spent large chunks of their life interpreting Hegel is considered just not to understand him well enough to see the valuable contributions he makes the claim that Hegel is worthwhile becomes unfalsifiable. Effectively it’s like saying that only priests are qualified to concluded God is unlikely to exist.

So some more real philosophy will start showing up here now that I’m back taking philosophy classes this term but for now some discussion on the practice of philosophy.

So it is quite possible I just have a misconception of applied bioethics but I keep hearing bioethicists on NPR or quoted in papers and what they do seems to have very little to do with philosophy. I’m sure their is real philosophy done by academic bioethicists but increasingly people called bioethicists seem to be in the pragmatic position of advising hospitals, scientific research and placed in other applied roles. Of course someone needs to make these decisions and for the most part the people who do bioethics seem to reach acceptable answers (in some sense) but there seems to be a fundamental tension between this pragmatic role and the job of a philosopher.

What I find particularly troubling is that the applied bioethics I hear about in the news and mentioned in scientific studies is so uncontroversial. Where, for instance, are the utilitarian bioethicists who are comfortable authorizing scientific studies which lack clear consent (the subjects are deceived in some way that goes beyond what is currently accepted) so long as the potential for benefit is great. Where are the Peter Singers of applied bioethics who view animal experimentation as only differing in degree from human experimentation. Alternatively where are the Rawlsians (or other theory if I am making a mistake about Rawls) who object to the rich paying for better doctors? Even if this positions are discussed in bioethical debates I very much doubt that any hospitals are stealing organs from non-donors because the expected harm from discovery was less than the benefit to the recipient. To cap it all off I just heard a bioethicist on NPR talking about avoiding genetically engineering pig genes into other products so as not to surreptitiously inflict pig on those who don’t eat pig for religious reasons. The justification he provided was not that it was in fact morally wrong to eat pig or even that we have a universal moral obligation to respect the desires of others (if this was the justification these people should be treated no differently than those who oppose genetic engineering for other grounds) but instead seemed to reflect the public sentiment that people’s religious beliefs are deserving of greater respect than other beliefs.

I don’t mean to criticize the people who are doing this job. It is probably better policy for hospitals and researchers to adopt some pragmatic generally inoffensive guidelines for behavior. However, their role is not really the discovery of truth or the development of philosophical arguments but rather the crystallization of public opinion with a grain of local consistency thrown in. Sure their ultimate goal is to achieve morally good ends but that is also the goal of politicians, humanitarian workers and virtually anyone who feels an obligation to be good.

In fact the applied nature of bioethics seems to make it impossible for them to both achieve moral ends and do philosophy. In order to be a philosopher I think at a minimum one must feel that the end product of your work should be the best argument possible for your position and view internal inconsistency to be a fatal flaw. However, the result of any serious attempt to create consistent moral theories is bound to run afoul of the public’s emotional commitments to conflicting principles (e.g. strong affirmation of the claim that all men are made equal combined with a strong aversion to considering the welfare of foreign nationals as equal to that of US citizens in law making). Sure any small number of these conflicts can be explained away but, as with every other interesting area of philosophy, the results of a serious attempt to work out a consistent theory are going to be counterintuitive in some respects. In fact, if we think the results of philosophical moral theory are only going to contradict intuition in trivial ways we should give up on unapplied moral philosophy as uninteresting. Yet, for pragmatic reasons and to achieve other larger moral goals I doubt applied bioethicists can be giving advice of the kind, “Yes if no one were to find out it would be better to steal organs from non-donors but the risk of discovery makes it better not to do it.”

Ultimately, it seems that the role of bioethicists is more akin to that of a judge than that of a philosopher. Like a judge an applied bioethicist cannot and should not substitute his beliefs about what moral positions are truly justified for the underlying values set by the public. Like a judge it is important that bioethicists come to some kind of common agreement, i.e., follow precedent, not figure out new clever arguments showing the current system is subtly inconsistent. Finally, and most strikingly, it seems appropriate for a bioethicist to be offered jobs on the basis of having mainstream or reasonable views (e.g. not really advocating utilitarianism) in a way which would be an egregious violation of academic freedom and a disservice to truth if they were truly a philosopher (of course philosophers need not be employed by a university but it seems in some sense essential that they have great freedom to take unpopular opinions).

Of course one might just grant that applied bioethicists aren’t philosophers in the traditional sense and take this step in stride. However, I am worried (though not convinced) about this for two reasons. First it only adds to the confusion in the public about what philosophy really is not to mention creating confusion in academic environments. Secondly, and more importantly, it appears to create the perception that applied bioethicists somehow have the ability to divine what is truly right and tell that to the public instead of merely interpreting what the public decides is acceptable. However, it is quite possible that my concerns merely stem from an inaccurate perception and I would appreciate feedback from anyone who knows more about the matter than I do. Especially if you have done any teaching in this area. (My comment system is a bit flaky so hit reload after you post to see if your comment is there)

Lately it seems the criticism that philosophy is too combative pops up everywhere. Most disturbingly in the form of allegations of implicit discrimination against women. Weirdly even when these criticisms come from philosophers they focus on whether people find the combative nature of philosophy a turn off. However, as I was reminded by this very nice post over at logicandlanguage.net reminded me (and inspired me to finally write this up) many normal academic interactions can be quite difficult for many people. Some individuals find lecturing terrifying, others (like myself) hate editing our papers (and our blog posts) and other people dislike the combative nature of philosophy. Clearly just because some people find lecturing unpleasant doesn’t mean we should stop requiring professors to lecture, holding lectures is quite central to the philosophical endeavor. Thus the important question about combativeness is whether it is also important to the philosophical endeavor and thus just another valid requirement for the discipline some people will find unpleasant. (more…)