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Here are some favourite lines from Alexander Pope’s Essay on Man.

Know, then, thyself, presume not God to scan;
The proper study of mankind is man.
Placed on this isthmus of a middle state,
A being darkly wise, and rudely great:
With too much knowledge for the sceptic side,
With too much weakness for the stoic’s pride,
He hangs between; in doubt to act, or rest;
In doubt to deem himself a god, or beast;
In doubt his mind or body to prefer;
Born but to die, and reasoning but to err;

Learn each small people’s genius, policies,
The ant’s republic, and the realm of bees;
How those in common all their wealth bestow,
And anarchy without confusion know;

Slave to no sect, who takes no private road,
But looks through Nature up to Nature’s God;

A current | LinkedIn discussion asks “Did man invent Maths or was it there all along waiting to be discovered?”. So I thought I’d take the return of that old question as a prompt to review and express some thoughts on the nature of the subject, of “truth” in general, (and of us).

In fact Elias Gourtsoyannis’ description of the views Lakoff and Nunez (which I guess I should make a point of reading!) sounds very like my own feelings on this issue – which also echo my (possibly not entirely faithful) reading of Alexander Pope’s “The proper study of mankind is man”.

As I see it, the rules of logic and axioms of set theory (from which all else in conventional mathematics can be constructed) are really as much expressions of how our minds work as of any external “reality” with which we have to deal.

And, as Colin says, when we find interesting new new tautologies to add to the body of mathematics (which consists of nothing else after all), the question of whether we are inventing or discovering is really just about the attitude with which we do so.

Of course, the reference to set theory as a foundation is merely an example. But most of the body of any mathematical system consists of tautological consequences of some small set of propositions which are taken as axioms.

I think that the part of mathematical activity which builds an edifice of theory on top of its axiomatic foundation is (contra the building metaphor) actually more a voyage of discovery than an act of invention. After all, once the axioms and rules are set, we have no choice to invent anything but what is already there.

But a claim for invention can surely be made about the axioms, and I believe, also about the rules of logic.

Apostolos Koutropoulos asks whether blogging or participation in discussion forums is preferable for participants in a MOOC. But since it’s a while since I participated in any course with him I would not have seen his question (or other stuff he has written in the interim) if it had been locked away in a course-specific discussion forum.

My own preference (expressed in CCK11) is for people to use open searchable/feedable channels for public or semi-public communication with trackback/pingback to enable the following of conversational threads where everyone is in control of access to their own content.

Unfortunately, take-up of this idea seems to be hampered by a perception that it is harder than it is, and the number of active independent voices is swamped by those who choose closed and limited systems like Facebook and LinkedIn.

But I am not ready to give up yet. And in the meantime am quite happy here talking to myself.

Nick Hanauer’s TED Presentation About Why Rich People Aren't Job Creators was initially withheld from publication. So now it’s up to others to make sure the message keeps getting repeated.

Wealth concentration reduces total consumption (because the wealthy cannot actually consume as high a multiple as they earn and so just buy land and hoard capital rather than creating jobs by consuming goods and services).

Philosophy seems to have a self-esteem problem these days. Partly it is a matter of depending for a sense of self-worth unduly on the opinions of others, but there is no denying that some of those opinions are negative. The Sokal event was in response to a particular school of apparent nonsense, but we also have the earlier remarks of Feynman on philosophy of science, and more recently Hawking on the relevance of the subject as a whole.

And now we also have the nasty dust-up over Lawrence Kraus’s childish and dismissive response to David Albert’s review of the title of his book. Gary Gutting sensibly asks Can Physics and Philosophy Get Along? (in the NYTimes on May 10), and provides a reasoned discussion of the sources of disagreement, but I think there is a bigger issue.

Perhaps it might be worthwhile for philosophers to ask if there is some good reason for the currently perceived disrespect of philosophy rather than just going into defense mode – and for those who do so ask, I have a suggested partial answer.

To me the value of Philosophy lies not in providing answers but in asking questions, and the problem as I see it is that many in the discipline present themselves as having some special kind of expertise in giving answers or resolving problems. Even when the claim of expertise is made more for the analysis and histories of proposed solutions (as opposed to their actual finding and evaluation) I still think that the emphasis on solutions is problematic.

A couple of my favourite examples of bad philosophising may help to make the point. Although Searle’s ‘Chinese Room‘ is mere foolishness as a “refutation” of “strong AI”, it may perhaps serve as a useful source of questions to clarify what the proponents of strong AI are actually saying. And similarly the “Gettier Problems” can lead to clarifying questions regarding the intent of those who “define” knowledge as justified true belief. In both cases a question might occasionally lead the hearer to identify something they had actually overlooked, but it remains open to the more likely possibility that the intent of the “folk” was actually much more sophisticated than the philosopher had understood it to be.

This time it’s christians taking offense (and getting support from the legal system in India) at the”blasphemy” of revealing the actual mechanism behind a purported “miracle”.

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Robert Reich’s ‘Thoughts on Tax Day 2012’ is worth noting if only for its reminder of two famous quotes.

Oliver Wendell Holmes, Jr.(1904): “taxes are the price we pay for a civilized society.”

Louis Brandeis (1897):  “we may have a democracy or we may have great wealth concentrated in the hands of a few, but we cannot have both.”

One reason for believing the latter was attempted in a much more recent quote.

Abbott Joseph Liebling (1960): “Freedom of the press is guaranteed only to those who own one.”

But this actually isn’t quite right. The cost of publication was never all that high and now it is negligible. And so it’s not the press but the audience that is hard or costly to obtain.

The reason Brandeis was right is because those with great wealth are better placed to buy or bribe the attention of voters to their message.

And the situation is unstable. Once a small class acquires more than half the wealth, that class has the power to buy more than half of the voters’ attention and with the not unreasonable assumption of total gullibility (ie every viewer is immediately persuaded by the last message heard) that is enough for them to further cement their position. And even without total gullibility it may still be possible to persuade the majority with sufficiently repeated exposure so a high enough domination of the economy may be sufficient to control the politics so as to be self-perpetuating.

Byron Jennings of TRIUMF has a blog at Quantum Diaries where his latest post challenges the Intelligent Design crowd to actually make some kind of testable prediction.

An alternative to making predictions, though, is just to declare the opposition in default for failure of postdiction and that is what  actually seems to be the preferred strategy of creationists. As Jennings says, “Being able to describe past observations is just the price to play the game, and with sufficient ingenuity, can usually be done.” Yes, and I am pretty sure that natural selection from random variations can in fact do the job. But given the effectively infinite variety of life, the task of explaining all past observations will never be done. When we have explained the eye that sees, then there’s the eye on the peacock’s tail, and after that the I of conscious experience, and then who knows what. If we don’t want to appeal to magic then the price of this game will never have been paid in full.  Of course finding the price of admission then becomes a game in itself, and we should thank those of little ingenuity whenever they come up with interesting puzzles for us to solve. (Yes, we have usually thought of whatever they suggest long ago, but we should still thank them out of politeness – and then ask them to go out and find us more challenging problems to solve.)

British Chief Rabbi Lord Sacks claims to know The Limits of Secularism, but he seems to be confusing secularism with science rather than just considering it as freedom from religion.

The two essential roles that he reserves for religion are the answering of big questions and the support of community and fellow-feeling. But he seems unconcerned as to whether the purported answers are in fact true, and is blind to the way that faiths which unite their adherents divide them from others.

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Law professor Stanley Fish probably knows quite a bit about evidence, but from his recent article with the above title I am led to doubt that he really understands much about science.

In particular, his main point appears to be based on a misunderstanding, for he says:

What I do assert is that with respect to a single demand — the demand that the methodological procedures of an enterprise be tethered to the world of fact in a manner unmediated by assumptions — science and religion are in the same condition of not being able to meet it (as are history, anthropology, political science, sociology, psychology and all the rest).

When a scientist expresses the criteria of an experimental test in terms of the theory being tested, that is only a shorthand for those familiar with the theoretical context and the true test can always be expressed in terms that require no theory-specific assumptions. For example the prediction that “this collision will produce an output of that particle” is just shorthand for something that could be expressed (though at much greater length) in terms of statements like “if you set those dials this way then that needle will point to this mark”.
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Philosophers seem to always be worried about not being taken seriously as providers of answers to important questions. But Philosophy Is Not a Science and so it is foolish of them to expect that kind of respect. They do not provide answers but they do often provide useful (albeit arguably often meaningless) questions and/or commentary which challenge the presumptions of our language and are better interpreted as art intended to influence our mood and mindset than as providing any kind of authoritative answers to real questions.
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I guess it’s at least a way of generating traffic for The Chronicle Review and those who have written books and articles on the subject, but I don’t know if there’s anything really new being said – either in those articles or in the responses from Russell Blackford and others.

But I do generally seem to agree with Blackford on most things and no less so this time when he says “I’m coming to think increasingly that all this talk of ‘free will’ isn’t very helpful for understanding our actual situation, partly because we don’t seem to have much clarity or agreement about what it even means”.

I also like Blackford’s apparent suggestion in his second response to Jerry Coyne that determinism is to some extent a *requirement* for free will. If what I actually choose to do or think at any point is not a function of the immediately preceding mental state which I have by previous cogitation brought into being, then that indeterminacy denies rather than grants me the capacity to control my own mental destiny.

This is along the same lines as my own claim that determinism is a requirement for responsibility in that the idea of responsibility is based on the capacity for being influenced by a response to our actions (or by observation of responses applied to others) in such a way as to perhaps modify our future behaviour.

What Is Science? From Feynman to Sagan to Curie, an Omnibus of Definitions is collected by Maria Popova at Brain Pickings,

and John S Wilkins at Evolving Thoughts has The Knight’s Song, or What is a Theory?

My own take is that a science is any teachable art of making correct testable predictions. With any restriction as to method or attitude being superfluous to the definition (though not without compelling interest in practice).

And I would say that a theory is just any specific process for making such predictions – including explanatory theories (such as “the butler did it”), theories predicting the results of further investigation (such as “you will find the money hidden in his room”), and mathematical theories such as analytic function theory or analytic number theory as being respectively the body of techniques of predicting and proving results a specific area or the possible results from  application of a specific class of methods.

William M. Briggs » Good Friday: Rally For Reason With St Anselm’s Ontological Argument believes that one can “come to religion through rational argument”.

Of course, if one admits a wrong argument expressed in a superficially rational-looking form as a rational argument then one can come to anything through rational argument, but the claim to rationality depends on a willingness to see and acknowledge flaws in one’s argument.

It is irrational to persist in believing arguments which have been shown to be incorrect, and it is only slightly less irrational to repeatedly fall for false arguments in favour of a proposition just because you want to have a “reason” to believe it.

Frankly, the only faith for which I have any respect is that which admits it is *not* supported by reason. If the existence of God were provable by reason then there would be no reason for faith and so to claim to have made an act of willful faith would itself be unreasonable. (Except perhaps if belief in god were attributed to faith in reason, which then sets reason above God – which may well be a greater sin than not believing in “him” at all.)

P.S. While it is not “irrational” to make a logical error it is arguably stupid to do so – and since (at least in my experience) we all make stupid mistakes quite frequently, I suspect that those who are offended by the charge of stupidity are displaying a higher level of arrogance than that of which they accuse their accusers.

Russell Blackford seems to like the fact that a recent Pew survey shows more Americans are getting wary of religion in politics.

So do I. I am offended by the view that religions should have privileged treatment under the law and I disagree strongly with many of their other views on how society should be run. So I am happy to see any move towards defeating those views.

But I am not so keen on the wording which suggests that people are prepared to say that churches should not express those views.

I am happy to see that Blackford, too, is not actually opposed to the churches expressing views on political matters, but I don’t understand why he thinks their expressed views should not be based on theological considerations. Shouldn’t they have the same right as any other group to express views that are ill-founded and wrong?

This ‘debate’ at The Chronicle Review came to my attention via Jean Kazez and Russell Blackford.

The fact that Facebook Co-Founder Chris Hughes has purchased The New Republic is a good sign that those who bewail the impact of the web on serious writing.

This New Yorker article gives a nice blend of personal,  scientific,  and species history. It would be interesting to know if what we picked up from our bigger brained but ultimately less successful cousins was just random or if it includes anything useful.

And as to the question of how “we” won, I am still inclined to speculate that there’s some kind of tie in between our capacity to learn from and cooperate with peers, a willingness to take direction from others (sometimes “blindly” so), and the tendency towards art and decoration – especially that of our own bodies to help identify “team” members in conflict situations. But now it looks as if the competitive team spirit and blind obedience to authority that won us the world may be about to become our undoing.  It would be ironic if some admixture from the Neanderthals were to be what we need in order not to destroy ourselves in religious conflict.

Some minds are changed a bit, and I think for the better, by this LinkedIn discussion, but it also includes a good example of exactly what I was concerned about in my comment on Keith Devlin’s original piece on the topic.

Here’s the link:

Should kids be told that multiplication is repeated addition? | LinkedIn.

First the good.

The question posed is a real paedagogical question to which I don’t think we know the answer, and the discussion led to some interesting examples of ways that the scaling interpretation of real number multiplication might be conveyed to small children . I don’t think it’s yet clear how early they can truly identify and compare different geometric magnitudes  (viz experiments comparing volumes for example) but this is certainly a worthwhile area for research.

But now the bad.

Part of what turns people against mathematics is the certainty of its results which means that one’s mistakes are hard to hide.  But what’s even worse than being found indisputably wrong when wrong is being told with a claimed mathematical level of authority that one is wrong when one cannot understand the argument.  And I think Devlin’s campaign against “repeated addition” makes it more likely for children to be exposed to that kind of dogmatic smackdown.

My objection to Devlin is that he claims as mathematical fact positions which are essentially philosophical. He makes no attempt to show any inconsistency in the  definition of multiplication of natural numbers by a recursive algorithm of repeated addition, but just claims in his first piece that this doesn’t capture what might be called the “essence” or “real meaning” of the corresponding operation on the reals and in his most recent effort he makes the claim that recursion is not a case of repetition.  While not mathematically provable, either of these may indeed be a reasonable philosophical position (though I don’t subscribe to either of them). And they may have valid paedagogical implications (which I might agree with despite not agreeing with his philosophy).  But I have always suspected that the way he said it would encourage people to dogmatically insist that repeated addition is  “not multiplication. Not in any sense; not even on the naturals”. And unfortunately there was an instance demonstrating just that in the course of the LinkedIn discussion.

My fear is that teachers who are a bit insecure about their understanding of mathematics will be led by such attitudes to undermine rather than support the understanding of a child who, on being introduced to multiplication says “Hey, I get it. This is just repeated addition!”

When that happens, I have no doubt that the teacher’s response should be an enthusiastic “Yes!”,  and it should come with no anxious look or subsequent undermining “but…”

It is perfectly fine to follow the “Yes!” with some thing like “and here are some things we can use it for..”  (see the wonderful poster by Maria Droujkova for some nice examples) . And of course these may include, as Maria does, examples which help to prime the student for seeing things another way. But it is up to the student to eventually see the next step, not for the teacher to immediately suggest that she has somehow fallen short.