I've been looking into origami-based geometry recently, on and off, and it's quite a fascinating subject. Geometry is based on constructions (it is the use of constructions that led C. S. Peirce to note that geometrical arguments can draw conclusions that strictly speaking go beyond the premises by, so to speak, abstract experiments with what may be but need not be). Our geometry developed by using three basic kinds of construction devices:
straightedge (or ruler)
compass
neusis (i.e., markings on a straightedge)
Neusis was known to the Greeks but usually depreciated by them; it was often regarded by them as insufficiently noble and abstract. And, indeed, the rejection of neusis shows that they were more interested in the abstract than applications, because neusis allows you to prove things that mere straightedge and compass don't. Regardless, Euclidean geometry is based wholly on straightedge and compass, and, indeed, can be regarded as a general theory of straightedge-and-compass construction: Euclid's geometry starts with a set of basic constructions with straightedge and compass that you are allowed, and then builds everything from there.
The reason for these particular instruments of construction is due to the way the ancient Greeks did geometry -- stick in the sand, chalk on slate, or what have you. It would be entirely possible, however, to do geometry with nothing but folded paper. Hence origami geometry, which has been studied at some length in the past two or three decades. It turns out that origami geometry allows for some very powerful mathematics. There are seven basic constructions possible using folds, which become the axioms of origami geometry; these axioms, having been first discovered by Jacques Justin, are generally known as Huzita-Hatori axioms:
(1) Given two points, there is a unique fold passing through both.
(2) Given points p1 and p2, there is a unique fold that leaves p1 on top of p2.
(3) Given lines l1 and l2, there is a fold that leaves l1 on top of l2.
(4) Given a point p1 and a line l1 there is a fold perpendicular to l1 that passes through p1.
You can do a fair amount with these four alone, but the geometry is weaker than Euclidean geometry. To beef it up a bit, you can add:
(5) Given points p1 and p2, there is a fold passing through p2 that leaves p1 on top of l1.
With these five constructions you can do everything that you can do with straightedge and compass. We can take it another step, though:
(6) Given points p1 and p2 and lines l1 and l2, there is a fold that leaves p1 on l1 and p2 on l2.
This is a neusis axiom, and when this construction is allowed, you can prove things that can't be proven by Euclid. The seventh axiom doesn't actually add anything new; but it's often thrown in because it is still one of the seven possible basic constructions:
(7) Given a point p1 and lines l1 and l2, there is a fold perpendicular to l2 that leaves p1 on l1.
Algebraically speaking, you can think of straightedge-only constructions as powerful enough to prove things describable as linear equations. Adding the compass lets you do quadratic equations generally (in effect, the compass lets you find the geometrical equivalent of square roots in a consistent way). Neusis allows you to do cubic equations (in effect, the compass lets you find the geometrical equivalent of cube roots in a consistent way). Quartic equations are based on square roots of square roots, but quintic equations and so forth are beyond any of these construction methods to achieve. However, apparently, by adding to the Huzita-Hatori axioms so that more complicated constructions are allowed, you can start solving quintic equations and even higher, thus stretching geometry to cover a little more. (There's bound to be some stick-in-sand equivalent of these complicated multifolds, but I haven't ever come across anyone who identified what it would be, and I don't have the mathematical ability myself to guess at what would be required beyond straightedge, compass, and neusis to quintisect an angle.)
There is a world of mathematics in a little folded paper.
Friday, June 29, 2012
Thursday, June 28, 2012
Southworth
Yesterday was the feast day of St. John Southworth, although today is the anniversary of his martyrdom. As you would guess from the name, he was an English martyr. He studied at Douai to become a priest; when he returned to England he was caught and condemned to death in 1627, but after three years in prison the sentence was commuted to deportation. He later returned to England, and settled quietly in Clerkenwell. While he was there a plague broke out and he devoted himself to tending to the sick. This was during the Interregnum, under Oliver Cromwell as Lord Protector. Unfortunately, this brought him to the notice of the authorities. He was arrested, let go, and then arrested again, at which point he was tried for the capital crime of being a Catholic priest, and of course he was convicted, in part because he insisted on pleading guilty to the charge of being a priest. He was then sent to Tyburn. It was a rainy, stormy day, but executions at Tyburn often drew large crowds regardless of weather and this was no exception. As sometimes happened, he was allowed to give a speech at the gallows, and we actually have an eyewitness account of the speech. Christ sent the apostles, Fr. Southworth said, and the apostles their successors, and the successors had sent him; and he had done as he had bid. He intended to no evil against the Lord Protector, and only sought to save souls:
(You can read the full speech attributed to Southworth here.) At this point he was pretty much told to hurry it up, so he ended with prayer and was hanged, drawn, and quartered along with a handful of counterfeiters; he was about sixty-two years old. If I recall correctly, he was the last person in England to be executed for being a priest until the Popish Plot scare stirred up by Titus Oates (the anti-priest law continued to exist but was not enforced for quite some time), and is the only English Catholic martyr under Cromwell (most Catholic martyrs under Cromwell were Irish).
How justly, then, I die, let them look to who have condemned me. It is sufficient for me that it is God's will: I plead not for myself, (I came hither to suffer,) but for you poor persecuted catholics, whom I leave behind me. Heretofore liberty of conscience was pretended as a cause of war; and it was held a reasonable proposition that all the natives should enjoy it, who" should be found to behave themselves as obedient and true subjects. This being so, why should their conscientious acting and governing themselves according to the faith received from their ancestors, involve them more than all the rest in that universal guilt? which conscientiousness is the very reason that clears others, and renders them innocent. It has pleased God to take the sword out of the king's hand, and put it in the protector's. Let him remember that he is to administer justice indifferently, and without exception of persons. For there is no exception of persons with God, whom we ought to resemble. If any catholics work against the present government, let them suffer; but why should all the rest who are guitless, (unless conscience be their guilt,) be made partakers in the promiscuous punishment with the greatest malefactors?
(You can read the full speech attributed to Southworth here.) At this point he was pretty much told to hurry it up, so he ended with prayer and was hanged, drawn, and quartered along with a handful of counterfeiters; he was about sixty-two years old. If I recall correctly, he was the last person in England to be executed for being a priest until the Popish Plot scare stirred up by Titus Oates (the anti-priest law continued to exist but was not enforced for quite some time), and is the only English Catholic martyr under Cromwell (most Catholic martyrs under Cromwell were Irish).
Wednesday, June 27, 2012
Ominous
An interesting recent ruling from Germany:
(Circumcision is still possible; the decision simply sets a precedent according to which those who perform it are more likely to be prosecuted.) I read somewhere something praising this decision because the court showed that it did not care that the decision might be regarded as anti-Semitic. It's certainly true that no court that was afraid to look anti-Semitic could render this decision. I'm not convinced that German courts suggesting that Judaism and Islam are contrary to fundamental human rights is an admirable thing, though.
ADDED LATER: Samuel Goldman puts his finger on a key part of the problem here:
ADDED LATER II: Gilbert at "The Last Conformer" discusses the matter more specifically. (ht)
Germany’s Jewish and Muslim communities responded with outrage today to a highly controversial court ruling which stipulated that the circumcision of young boys on religious grounds caused bodily harm and infringed a child’s right to physical integrity.
The divisive verdict, delivered by an appeals court in Cologne, involved the case of a Muslim boy who became seriously ill after undergoing the procedure, and ruled that the individual rights of the child took legal precedence over the religious rights of its parents.
The judges concluded that “circumcision contravenes the interests of the child to decide later on in life about his religious beliefs” and “the fundamental right of the child to bodily integrity outweighs the fundamental rights of the parents”.
Male circumcision is not illegal in Germany, but until the court ruling it had inhabited a legal ‘grey area’ which had allowed doctors to carry out the operation. Some medical organisations advise doctors to draw up a contract with their patients before performing the operation to guard against possible legal action.
(Circumcision is still possible; the decision simply sets a precedent according to which those who perform it are more likely to be prosecuted.) I read somewhere something praising this decision because the court showed that it did not care that the decision might be regarded as anti-Semitic. It's certainly true that no court that was afraid to look anti-Semitic could render this decision. I'm not convinced that German courts suggesting that Judaism and Islam are contrary to fundamental human rights is an admirable thing, though.
ADDED LATER: Samuel Goldman puts his finger on a key part of the problem here:
There’s no indication of specific hostility to Judaism here. Nevertheless, the ruling is the logical consequence of a concept of religion implied by Protestantism and articulated philosophically by Benedict Spinoza and John Locke.
According to that view, religion is rooted in private belief. Associations and rituals are legitimate only to the extent that they are submitted to voluntarily by consenting adults, who can withdraw their consent at any time. And religious obligations can never trump the civil law.
There are good reasons that this position was appealing in the 17th and 18th centuries. Trouble is, we’ve forgotten not only that it doesn’t fit many older traditions, including Judaism and Roman Catholicism, but that it was specifically designed to exclude them. The understanding of religion’s legitimate sphere that informed the Cologne court’s ruling, in other words, is not theologico-politically neutral. It was, and remains, a polemical concept that elevates state over church, individual over community, consent over continuity in ways that traditional Catholics and Jews find hard to accept.
ADDED LATER II: Gilbert at "The Last Conformer" discusses the matter more specifically. (ht)
Magical Thinking with Arguments
Gilson somewhere notes that the tendency of people in the nineteenth century to assume that Newtonian physics showed that the universe was deterministic was not based on any kind of rigorous argument, and could not be. What it actually was based on was something more like an imaginative sense of analogy based on how it was often presented. And this seems quite right. Actually moving from Newtonianism to a deterministic universe requires either ignoring or rejecting any number of possible qualifications (e.g., do we have good reason to think that Newtonian physics actually covered everything in the universe). In fact, it's at best controversial whether Newtonian physics is even fully deterministic in its own right; John Norton's dome is a famous case in which a scenario apparently consistent with all basic principles of Newtonian physics yields a result that violates determinism. But when we think of Newtonian physics we think of things like billiard balls hitting each other, and thus a billiard ball universe, etc.
Nineteenth- and early twentieth-century anthropologists often covered this kind of reasoning with the label 'magical thinking' -- or did when they found it in non-European cultures. But obviously this is not something that is unique to the 'primitives', except in the sense that everyone is primitive. And I've actually been considering the idea recently that people's evaluations of arguments are strongly affected by the kinds of associative similarity that anthropologists looked at when discussing magical thinking. In general, all their explanations of it treated the main nub of magical thinking as not making the distinction between associations between ideas and relations between things represented by ideas. It's certainly the case that people often fail to do this in evaluating reasoning, confusing associations among their own ideas of arguments and positions with actual relations among those arguments and positions. Frazer famously argued that sympathetic magic could be summarized in two laws:
Contagion: Things in contact continue to be in contact even where physical contact is broken.
Similarity: Like effects can be had by making causes alike.
Frazer, of course, was assuming that we are talking about physical objects, but one can easily find analogues of these in people's evaluation of reasoning. For instance, people have a tendency to assume that because two different philosophical positions have been found together at some point that they really have some connection, even if there is no logical requirement for this.
In fact, of course, as Levy-Bruhl pointed out, many of the instances of magical thinking were straightforwardly recognizable as logical fallacies -- post hoc ergo propter hoc was the one he focused on. But, of course, this is arguably committed as much by Europeans as by any 'primitive' tribe; we just think of it as bad magic. And this kind of thing can be found in evaluation of arguments and positions as well -- the 'genetic fallacy' is just a name for post hoc ergo propter hoc when we are talking about positions.
Likewise people have their rituals in argument evaluation. Philosophers like to set out the premises in an orderly numbered fashion, and tend to regard this as making an argument clear. Whether or not it actually does so depends; unless the argument is being made from scratch, this procedure involves rearrangement and interpretation, so whether it actually does make things more clear, in terms of increasing understanding, seems to vary considerably. But it still feels like you are bringing order and clarity to a disordered muddle, so you find people who will swear by it, even though it's not difficult to find cases where it clearly introduced a distortion. There's an argument to be made -- it would, of course, be controversial among those who engage in this kind of practice -- that such people are taking the ritual itself to be a kind of clarity, by sympathetic magic, and are taking arguments in this form to be better arguments simply because they conform to ritual expectation. It may even have good practical results, if so; a ritual might well put one in the right state of mind for a certain kind of work, and there's no reason to think that philosophical thinking doesn't sometimes need 'being in the right state of mind' as much as any difficult endeavor. And, of course, you find people who try to refute arguments by naming them -- a practice difficult to avoid, but not really all that different from shamans casting out illness by naming it.
Of course, unlike the anthropologists back then, we needn't regard 'magical thinking' as necessarily a bad thing; very likely it's just a first thing. It's rather funny reading Frazer on similarity, for instance, given that his account of the law of similarity for sympathetic magic often reads like a straightforward summary of Humean accounts of causation (which, of course, are based on associative similarities of a particular kind). It does not follow automatically from this that they are mere mumbo-jumbo. And something that may be poor reasoning in one context may be quite good in another context -- the failure to recognize this as at least a possibility is one of the most rookie mistakes when it comes to evaluating reasoning. And I'm often amused when anthropologists talk about how magical thinking often involves thinking that one's words have a causal influence on the world, since they often are sloppy enough that their formulation implies that getting someone to pass the salt by saying please is magic. Which perhaps it is; it's not as if any of us knows the precise causal connection, or even usually cares. In any case, my point here is that one need not take the label as an insult. What 'magical thinking' really is, of course, is a system of analogical inferences, and not only can we not avoid making analogical inferences, many such are quite reasonable. To that extent it's both unsurprising that people would engage in magical thinking when evaluating arguments and not necessarily in a bad way -- their 'magical thinking' might well provide a good starting point for rigorous analyses, although, of course, that might depend on any number of factors. It bears thinking, anyway.
Nineteenth- and early twentieth-century anthropologists often covered this kind of reasoning with the label 'magical thinking' -- or did when they found it in non-European cultures. But obviously this is not something that is unique to the 'primitives', except in the sense that everyone is primitive. And I've actually been considering the idea recently that people's evaluations of arguments are strongly affected by the kinds of associative similarity that anthropologists looked at when discussing magical thinking. In general, all their explanations of it treated the main nub of magical thinking as not making the distinction between associations between ideas and relations between things represented by ideas. It's certainly the case that people often fail to do this in evaluating reasoning, confusing associations among their own ideas of arguments and positions with actual relations among those arguments and positions. Frazer famously argued that sympathetic magic could be summarized in two laws:
Contagion: Things in contact continue to be in contact even where physical contact is broken.
Similarity: Like effects can be had by making causes alike.
Frazer, of course, was assuming that we are talking about physical objects, but one can easily find analogues of these in people's evaluation of reasoning. For instance, people have a tendency to assume that because two different philosophical positions have been found together at some point that they really have some connection, even if there is no logical requirement for this.
In fact, of course, as Levy-Bruhl pointed out, many of the instances of magical thinking were straightforwardly recognizable as logical fallacies -- post hoc ergo propter hoc was the one he focused on. But, of course, this is arguably committed as much by Europeans as by any 'primitive' tribe; we just think of it as bad magic. And this kind of thing can be found in evaluation of arguments and positions as well -- the 'genetic fallacy' is just a name for post hoc ergo propter hoc when we are talking about positions.
Likewise people have their rituals in argument evaluation. Philosophers like to set out the premises in an orderly numbered fashion, and tend to regard this as making an argument clear. Whether or not it actually does so depends; unless the argument is being made from scratch, this procedure involves rearrangement and interpretation, so whether it actually does make things more clear, in terms of increasing understanding, seems to vary considerably. But it still feels like you are bringing order and clarity to a disordered muddle, so you find people who will swear by it, even though it's not difficult to find cases where it clearly introduced a distortion. There's an argument to be made -- it would, of course, be controversial among those who engage in this kind of practice -- that such people are taking the ritual itself to be a kind of clarity, by sympathetic magic, and are taking arguments in this form to be better arguments simply because they conform to ritual expectation. It may even have good practical results, if so; a ritual might well put one in the right state of mind for a certain kind of work, and there's no reason to think that philosophical thinking doesn't sometimes need 'being in the right state of mind' as much as any difficult endeavor. And, of course, you find people who try to refute arguments by naming them -- a practice difficult to avoid, but not really all that different from shamans casting out illness by naming it.
Of course, unlike the anthropologists back then, we needn't regard 'magical thinking' as necessarily a bad thing; very likely it's just a first thing. It's rather funny reading Frazer on similarity, for instance, given that his account of the law of similarity for sympathetic magic often reads like a straightforward summary of Humean accounts of causation (which, of course, are based on associative similarities of a particular kind). It does not follow automatically from this that they are mere mumbo-jumbo. And something that may be poor reasoning in one context may be quite good in another context -- the failure to recognize this as at least a possibility is one of the most rookie mistakes when it comes to evaluating reasoning. And I'm often amused when anthropologists talk about how magical thinking often involves thinking that one's words have a causal influence on the world, since they often are sloppy enough that their formulation implies that getting someone to pass the salt by saying please is magic. Which perhaps it is; it's not as if any of us knows the precise causal connection, or even usually cares. In any case, my point here is that one need not take the label as an insult. What 'magical thinking' really is, of course, is a system of analogical inferences, and not only can we not avoid making analogical inferences, many such are quite reasonable. To that extent it's both unsurprising that people would engage in magical thinking when evaluating arguments and not necessarily in a bad way -- their 'magical thinking' might well provide a good starting point for rigorous analyses, although, of course, that might depend on any number of factors. It bears thinking, anyway.
Tuesday, June 26, 2012
JTB and Plato
Massimo Pigliucci recently committed one of my pet peeves:
Plato did not, in fact, maintain any such thing. In the Theaetetus, in which he discusses various possible accounts of knowledge, Plato's Socrates considers three possibilities: sensation, true judgment, true judgment with logos. Socrates rejects all three -- each one has problems. The major problem with the last (which is introduced not by Socrates but by Theaetetus as something he has heard elsewhere) is that no plausible account of a 'logos' actually gets us anywhere -- every way 'logos' is cashed out (Socrates tries out several different ways) 'true judgment with logos' ends up being either circular or obviously incomplete. That is, it always just ends up meaning 'true judgment that is knowledge' or it leaves something out, making knowledge indistinguishable from certain kinds of lucky opinion. Socrates concludes that the account is silly and a bunch of wind, and there the dialgue ends as Socrates heads out to his trial. The most plausible account of knowledge attributed to Plato himself, of course, is that knowledge is rational perception of the unchanging Forms.
JTB theories of knowledge are in fact quite minor accounts historically; Gettier's arguments against them have no purchase against Neoplatonist or Aristotelian accounts of knowledge, nor do they work against either empiricist or rationalist accounts of knowledge in the early modern period, and that covers the major accounts of knowledge in the history of Western philosophy. Like many of the things contemporary philosophers take as obvious, JTB does not have a very extensive history, and there is no clear point at which people picked it up because it was superior to any rivals. Indeed, as seen from the rather extensive misattributions of it to Plato, it often seems to be go with a complete ignorance of the fact that there are any rivals. But the history of philosophy consists almost entirely of its rivals. Most positions take knowledge to be some kind of perception or apprehension; Aristotelian accounts take it to be logical derivation from immediate principles, which themselves are directly apprehended. There's a good argument to be made that almost everyone prior to at least the nineteenth century would have taken "justified true belief" to be a nonstarter as an account of knowledge for reasons broadly similar to those found in the Theaetetus.
Plato famously maintained that knowledge is “justified true belief,” meaning that to claim the status of knowledge our beliefs (say, that the earth goes around the sun, rather than the other way around) have to be both true (to the extent this can actually be ascertained) and justified (i.e., we ought to be able to explain to others why we hold such beliefs, otherwise we are simply repeating the — possibly true — beliefs of someone else).
Plato did not, in fact, maintain any such thing. In the Theaetetus, in which he discusses various possible accounts of knowledge, Plato's Socrates considers three possibilities: sensation, true judgment, true judgment with logos. Socrates rejects all three -- each one has problems. The major problem with the last (which is introduced not by Socrates but by Theaetetus as something he has heard elsewhere) is that no plausible account of a 'logos' actually gets us anywhere -- every way 'logos' is cashed out (Socrates tries out several different ways) 'true judgment with logos' ends up being either circular or obviously incomplete. That is, it always just ends up meaning 'true judgment that is knowledge' or it leaves something out, making knowledge indistinguishable from certain kinds of lucky opinion. Socrates concludes that the account is silly and a bunch of wind, and there the dialgue ends as Socrates heads out to his trial. The most plausible account of knowledge attributed to Plato himself, of course, is that knowledge is rational perception of the unchanging Forms.
JTB theories of knowledge are in fact quite minor accounts historically; Gettier's arguments against them have no purchase against Neoplatonist or Aristotelian accounts of knowledge, nor do they work against either empiricist or rationalist accounts of knowledge in the early modern period, and that covers the major accounts of knowledge in the history of Western philosophy. Like many of the things contemporary philosophers take as obvious, JTB does not have a very extensive history, and there is no clear point at which people picked it up because it was superior to any rivals. Indeed, as seen from the rather extensive misattributions of it to Plato, it often seems to be go with a complete ignorance of the fact that there are any rivals. But the history of philosophy consists almost entirely of its rivals. Most positions take knowledge to be some kind of perception or apprehension; Aristotelian accounts take it to be logical derivation from immediate principles, which themselves are directly apprehended. There's a good argument to be made that almost everyone prior to at least the nineteenth century would have taken "justified true belief" to be a nonstarter as an account of knowledge for reasons broadly similar to those found in the Theaetetus.
Seldom Comes the Better
All great and sudden changes are dangerous to the body natural, but much more to the body politic. Time and custom beget reverence and admiration in the minds of all men: frequent alterations produce nothing but contempt. Break ice in one place, it will crack in more. Mountebanks, projectors, and innovators, always promise golden mountains, but their performance is seldom worth a cracked groat. The credulous ass in the fable believed, that the wolf (his counterfeit physician) would cure him of all his infirmities, and lost his skin for his labour. When the devil tempted our first parents, he assured them of a more happy estate than they had in Paradise: but what saith our common proverb, 'Seldom comes the better.'
John Bramhall, "The Serpent-Salve". Bramhall was the Anglican bishop of Armagh, and a famous polemicist. He is most famous for his wrangling with Thomas Hobbes on the topic of free will, which I am considering doing a series of posts on in July or August.
Thomas More to His Whole School
In some late reading for St. Thomas More's feast-day last week, I was delighted to come across the following letter from More to his children, which I originally discovered here, and which you can also find in this nineteenth century work at Google Books. More faced a problem that many fathers today face: his job took him away from his family for very extended periods of time. He had to be at court, and that meant that long stretches went by during which his children did not see him. More's way of handling the problem was to write them letters, and as you can see from the following, they were brilliant. He teases them, admonishes them, and makes clear that he loves them. I especially like the sentence about Lent, for two reasons -- it makes clear to his children (without being too clumsy about it) that, even though he is away, when they observe Lent they are doing it with their father, and he recommends Boethius's Consolation of Philosophy as proper Lenten reading, a recommendation of which I very greatly approve.
As this letter suggests, More took the education of his children very seriously. One of More's distinctive features is that he was perhaps the biggest advocate of women's education in his day. It was common at that time for men, and sometimes women, to be skeptical of whether women were capable of learning serious subjects. Queens and princesses could perhaps get away with dabbling in such things, but royalty is to be humored; and even then there were many who did not really take it seriously. More would have none of it, and made sure his own daughters received a full humanist education, and in the case of Margeret, his eldest, succeeded so well that he turned some skeptics around.
Thomas More to his whole school,—
See what a compendious salutation I have found, to save both time and paper, which would otherwise have been wasted in reciting the names of each one of you, and my labour would have been to no purpose, since, though each of you is dear to me by some special title, of which I could have omitted none in a set and formal salutation, no one is dearer to me by any title than each one of you by that of scholar. Your zeal for knowledge binds me to you almost more closely than the ties of blood. I rejoice that Mr. Drew has returned safe, for I was anxious, as you know, about him. If I did not love you so much I should be really envious of your happiness in having so many and such excellent tutors. But I think you have no longer any need of Mr. Nicholas, since you have learnt whatever he had to teach you about astronomy. I hear you are so far advanced in that science that you can not only point out the polar-star or the dog-star, or any of the constellations, but are able also—which requires a skilful and profound astrologer—among all those leading heavenly bodies, to distinguish the sun from the moon! Go forward, then, in that new and admirable science by which you ascend to the stars. But while you gaze on them assiduously, consider that this holy time of Lent warns you, and that beautiful and holy poem of Boetius keeps singing in your ears, to raise your mind also to heaven, lest the soul look downwards to the earth, after the manner of brutes, while the body looks upwards. Farewell, my dearest.
From Court, the 23rd March.
As this letter suggests, More took the education of his children very seriously. One of More's distinctive features is that he was perhaps the biggest advocate of women's education in his day. It was common at that time for men, and sometimes women, to be skeptical of whether women were capable of learning serious subjects. Queens and princesses could perhaps get away with dabbling in such things, but royalty is to be humored; and even then there were many who did not really take it seriously. More would have none of it, and made sure his own daughters received a full humanist education, and in the case of Margeret, his eldest, succeeded so well that he turned some skeptics around.
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