I’ve always sorta suspected that explaining mathematics to students in very concrete ways wasn’t really a shortcut but it was like being offered a magic lamp with a genie who will grant your wishes. You know something about it is too good to be true even if you don’t really know what it is. Anyway apparently now there is some evidence that my intuitions are correct.

Of course I don’t think this means one should never use analogies to concrete situations or that math must be presented as uninterpreted formal manipulation. Nor is it really true that I didn’t know why I found the teaching of math in a super concrete fashion troubling. Ultimately math is the use of abstraction to solve problems. The power of things like group theory, analysis, recursion theory or even calculus is the result of abstracting the relevant features from the motivating examples. This is just as true for the subjects we try and teach our students as it is for the subjects we study ourselves. For example, in order for students to understand (not merely memorize) how to take the derivative of an integral using the chain rule and fundamental theorem of calculus requires the students have abstracted away from the idea of functions as rules they can right down in terms of powers,ratios, trigonometric functions and exponentials and can accept a define integral they can’t compute as just another function.

Ultimately, interpreted correctly I don’t think this study really says anything that shouldn’t have already been obvious. It doesn’t say that concrete examples can’t help students grasp the abstract concepts being presented (they can) it just tells us that there is no shortcut to mathematical understanding. If you want students to actually understand math that means teaching them to understand the concepts abstractly and if you take the easy way out and encourage your students to understand mathematics in a particular concrete fashion instead of offering the concrete examples/analogies and encouraging them to move beyond it their understanding will suffer.

As a side note the entire discussion around this study is a perfect illustration of the fundamental confusion at the heart of mathematical education today. Framing the study as showing that concrete teaching methods fail because they aren’t as effective in teaching students to solve other concrete problems seems totally backwards to me. Frankly in the modern world being able to compute when too trains will pass or find the symbolic integral of sin(x) isn’t an important skill for most people. Go ask anyone over 40 who doesn’t work in a technical field to solve a system of two linear equations, integrate a function, or even multiply fractions and they aren’t likely to be able to do it. If the particular skills we pass on in math class were so important to everyone how has our civilization manage to flourish up till now? It might be sad, you might not like it but the strategies of asking a technical friend, using a calculator, or just not dealing with mathematical problems work pretty damn well for most people. On the other hand the ability to abstract away particular features of a class of examples is a skill virtually all professionals require.

Ideally we would stop blindly teaching the traditionally expected skill set and ask what we mean to accomplish. If we want to give people practical skills to use in their life we need to take a look at some empirical data about what if any mathematical education translates to practical benefit. You might think that everyone should be able to calculate with percents and fractions but if most people simply use the calculator on their cell phone then we have to consider the possibility that we’d be better off helping them practice computing in that way. Certainly some people need to learn how to do things like multiply fractions, compute integrals and so forth but that isn’t everyone. On the other hand if the reason to teach mathematics is to impart quantitative reasoning and the ability to reason abstractly then why the hell is most of our mathematical education blind computation?

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

As American consciousness of global warming has increased and a consensus that we need to do something about it has emerged my confidence that we will actual address the problem has waned. Fundamentally global warming is a scientific, engineering and economic problem which requires a solution on those levels. Indeed, dealing with domestic CO2 emissions isn’t that hard of a problem. A CO2 tax would be an easy and (relative to GDP) a fairly inexpensive way to solve the problem. Admittedly there is a real worry that this would drive industries to the third world where they would be subject to less stringent emissions controls but this is one situation where an appropriate use of tariffs could address. In my opinion an optimal solution would be to offer developing countries tariff free export to the US and other participating industrialized countries in return for imposing taxes on CO2 emissions.

Unfortunately greater environmental awareness doesn’t seem to have increased support for sane policies like this one jot. Just in the last few days McCain announced his proposal for a gas tax holiday. Lest you think that this is only a proposal that caters to the trucks and guns crowd consider the fact that no democratic candidates would dare to propose an increased gas tax for fear of the public backlash. It doesn’t seem to matter that such a tax could be made revenue neutral and could even favor the poor people since people respond viscerally to expensive gasoline.

Instead of responding rationally to the global warming issue people, especially those claiming to be environmentally conscious, instead lash out at conspicuous consumption. It is somehow considered a moral hazard to buy a gas guzzling car, take plane flights, run an air conditioner or engage in other activities that have a salient link the emissions. This of course ignores the fact that the money people save as a result of these various conservation measures goes into buying other products which themselves likely have a large carbon footprint. All the tips about how to save electricity/gas or reuse items instead of throwing them out are particularly silly. After all if I save gas that reduces the price for other consumers who may then use more. Of course these factors are likely not 1-1 but it illustrates the point that urging people to avoid activities that seem wasteful is not only a waste of utility (moral guilt doesn’t discriminate between the people for whom running their air conditioning isn’t a big deal and those who get great utility from it) but it isn’t a very effective way of accomplishing the goal.

Unfortunately this attitude that environmentalism is really about eliminating consumer excess seems to be on the lips of every environmental activist I meet lately. They remark about how we will look back on our wasteful product packaging and huge trucks as gross and wasteful. Well hopefully we will look back on them as being inefficient in terms of carbon but there is absolutely no reason we shouldn’t expect the future to eventually provide cheaper energy, more disposable cruft, more gadgets and less need for conspicuous recycling (automated sorting) while making a smaller environmental impact. Even if this isn’t possible surely we ought to aim to improve our standard of living while saving the environment.

I don’t know where this idea that environmentalism is some sort of personal virtue of frugality came from but it’s not only a bunch of bunk but it’s hurting the environment. Not only does this attitude alienate many people who might get on board with a more pragmatic engineering/economic approach to the environment but it also competes with real solutions. People are willing to make a limited amount of sacrifices and if you tell them that they are being good people for enduring daily inconveniences like turning off their AC or buying a car they don’t like as much they won’t be as eager to ’sacrifice’ again by voting up the gas tax.

Perhaps that title is a little over the top since I don’t mean all doctors but it captures the sense I got from a recent radio discussion on KQED (NPR station) about at home genetic testing. On that show two prominent doctors were warning about the dangers and harms of these unregulated kits for at home genetic testing and complaining that people were getting this information without the counseling of a medical professional or a clear understanding of what the information meant. Hopefully their demands for regulation went beyond what most doctors would support but it was my sense that they represented a view that a sizable fraction of the medical community would support.

Now there are real concerns that ought to be addressed by regulation. Any at home genetic test should be regulated to guard against fraudulent misinformation or scientifically unsupported medical advice. They should also be regulated to ensure that they give accurate results and report the chance of type I and type II error to the test taker. However, it seems self-evident to me that as citizens in a free country we ought to have the right to any information about our own genetic profile we wish to receive. So long as the test manufacturer does nothing besides inform the customer of what genetic variants they possess and what the current research suggests about these variants they should have the right to supply us that test.

The doctors on the program raised the valid concerns that people might very well not know how to react to the type of genetic information they receive. For instance they implied that individuals might be better off not being told they have a gene that boosts their change of cancer by 1.2 times since people have trouble understanding what a small effect this is and may then demand extra tests that increase their risk and burden the medical system. Well perhaps these individuals would be better off not knowing this information but is that valid grounds to legally deny it to them? I think not. The doctors kept insisting that we needed studies to determine what information improved public health and what didn’t which is important but in terms of regulation misses the broader point that public policy has broader goals than maximizing public health.

No doubt public health would increase if restaurants and supermarkets were only allowed to sell us food that our doctors approved but obviously we should have the right to buy the food of our choice. The fact that people generally have a poor understanding of the effect of food choice on their health is irrelevant. In the long term we tend to find that the harms to innovation, frustration at being told what to do, and the slippery slope towards further regulation that result from the restriction of freedom are worse than the more obvious immediate harms. In other words if our society believes freedom is so important that we protect racist and bigoted speech then surely we ought not to restrict people’s access to their own genetic information just because they might not understand it.

If I want to find out if I have gene BLAH I should have the legal right to do so and so long as the company offering me the test for this gene doesn’t misrepresent our certainty about the role of BLAH they should have the right to offer me such a test. Of course doctors should be free to discourage us from using such tests just as they are free to discourage us from using tobacco or eating junk food but the law shouldn’t require us to do what our doctor asks. So long as doctors don’t drag their feet on incorporating genetic information into their practices as quickly as the science allows I expect the percentage of people who insist on ignoring their doctor’s advice will remain quite low. Ultimately though, the right to obtain information about our own genetic makeup is so central to our freedom and liberty that nothing short of a plague level public health catastrophe would justify denying this freedom.

Hopefully I merely misinterpreted the doctors on this show and they weren’t proposing that we bar companies from offering genetic information unless it offers a net public health benefit. However, doctors are people too and it’s very easy for people to get caught up in their personal mission (especially when it’s something noble like serving the public health) and forget that this isn’t the only, or even primary, aim of public policy. Of course if you are a doctor and you see people coming into your office with information from genetic tests that you don’t know how to properly interpret and demanding what are likely harmful procedures your likely to want to clamp down on these tests. It’s a natural human reaction to want to control things that interfere with our mission but that just isn’t how we should handle them in a free society.

Update:

This has been broken in the most recent nightly builds. I think I’ll wait till things settle down before I try and figure out how to do it again.

So keeping track of the various comments I leave on blogs around the web has always been a challenge. For awhile I tried using the service coComment but It’s hard to believe how astonishingly bad supposedly professional web developers can be at creating simple javascript tools. I could either install their bloated extension that would run their code on every webpage I visited (which I think made a call back to coComment on any page that looked like it might be a blog) or run a crappy bookmarklet that didn’t work very well. I’m sure the service works for some people but it’s yet another example of a bad system created because the authors thought everyone would want to use it in exactly one way (or should use it that way). I’m much happier with co.mments but sometimes I just want to remember a page/post not place it in the list of discussions I’m following, keep track of several types of pages (discussions and neat products) or I just want these links to be conveniently accessible from my toolbar.

With Firefox 3’s places support I’ve finally found a great solution. Say I want a folder that lists the 15 most recent pages I bookmarked with the tag ‘comments’. First I need to have bookmarked at least one page with that tag at which point I find the id of the folder “comments” under the tag folder. I did this by using the sqlite manager plugin for firefox and opening the places.sqlite database in my profile folder. (I think it may also be possible to do this by opening bookmarks.postplaces.html). You’ll know that you’ve found the right entry if it has the same name as the tag you’re interested in and has a parent of 4 (it’s in the tags folder). Once you’ve found the id of the tag folder you are interested in create a new bookmark and enter the following for it’s location:

place:folder=ID&queryType=1&group=3&sort=4&applyOptionsToContainers=1&maxResults=15

Where you should replace ‘ID’ by the id of the folder for the tag in question. This will then create a smart folder that will display the 15 bookmarks that were most recently visited with the tag in question. You can investigate other options and work out other nifty queries to try using this post from the mozillazine forums. In partcular if you chage the sort from 4 to 12 it will instead list the 15 most recently bookmarked pages with the given tag.

Note that I found that I had to restart minefield in order for it to recognize any changes in the querystring. The list itself with update as you add new bookmarks with the appropriate tag but if you decide that it should display only the 10 most recent rather than 15 you may have to restart your browser for it to recognize the change.