I’ve recently become aware of an entertaining controversy in the world of mathematics. It seems a dissident faction of mathematicians is advocating the replacement of the mathematical constant π with the related constant τ = 2π. T he self-described ‘Tauists’ are conducting their campaign with a form of ha-ha-only-serious dry humor that gently mocks the conventions of the mathematical literature, but if we receive it only as satire, we risk missing some serious and interesting issues in play.
For background, you can read The Tau Manifesto and a rebuttal at The Pi Manifesto. I’ve read some of the source arguments these were constructed from, such as Bob Palais’s original “π is wrong! “article; having done so, I think the manifestos sum up the state of play pretty well.
I was at one time a mathematician with a serious interest in foundational issues; that is, I was interested in studying the axiomatic basis of mathematics itself. One relevant question the π vs. τ dispute raises is what we actually mean when we say that “π is an important mathematical constant”. Is there any sense in which we can say that π = τ/2 is “more special” than τ = 2π, or vice-versa?
Curiously, though the disputants sidle up to the second question in a sideways manner, I haven’t yet seen anyone tackle the first one directly. Perhaps this is because it hasn’t attracted a foundationalist’s attention yet; most of the mathematical community, so far, seems to dismiss the dispute as trivial. I don’t think it is, though, and even if I’m no longer officially a mathematical philosopher myself I can at least play one on the net.
The real meat of this dispute is in how we evaluate competing notations for the same mathematical system, and to approach that question we need an explicit theory of what mathematical notation does and what it is for. The π-vs.-τ dispute has legs exactly because mathematicians in general have only a loose and implicit theory of notation; while it leads to broadly shared intuitions most of the time, it does tend in edge cases like the π-vs.-τ dispute to collapse into personal esthetic evaluations that cause a lot of argumentative heat exactly because they can’t really be logically defended.
In fact it is exactly this collapse that the Tauists are gently parodying, even as they make a serious case for τ. But the Tau Manifesto gets caught up in specific arguments about its controversy enough that it only glances at the more general question: what makes one mathematical notation better than another?
We can start by noticing that mathematical notation has two broad functions. One is to facilitate computation; the other is to help mathematicians generate intuitions about its subject matter.
The first of these is relatively easy to think about. In the past, changes in notation have brought about dramatic improvements in ease of computation, leading sometimes to very large consequences in the real world. Perhaps the most dramatic example was the shift from Roman numerals to modern positional notation during the early Renaissance. This made arithmetic so much easier that every human endeavor in contact with it got revolutionized, leading to results as diverse as double-entry bookkeeping, open-ocean navigation, and (arguably) the invention of physics. In more recent times, the invention of tensor calculus in the late 1800s proved essential for helping Albert Einstein and others perform the essential computations of General Relativity Theory.
The second use, helping generate intuitions, is much less well understood. No mathematician doubts that expressive notation is like wings for the mathematical imagination; nor that a clumsy, poorly chosen notation is like hanging weights on it. But, as in Hollywood, nobody knows what will work for audiences until it’s tried. Our evaluations of “expressive” and “clumsy” can usually be only be made after the fact and in a relatively fuzzy way.
But the most important property of good notation serves both purposes. Good notation expresses complex ideas in a simple and regular way. And this is something we can actually formalize, because human brains being what they are, “simple” unpacks to “few enough symbols to fit in the brain’s working storage”. Short formulas with large consequences are the greatest achievements of both pure and applied mathematics.
This gives us a metric. Suppose we have a list of theorems and derivations that we consider important, and two alternative notations for expressing them. There is a known way to map without loss from one notation to the other and back. Which, then, is better?
The simplest answer is, I think, the fundamentally correct one. Write them all down in both notations and count symbols. The notation with the lower symbol count wins, and not by accident but because handling it will impose lower overhead on the user.
The Tauists and pi partisans understand this well enough that they argue back and forth partly by listing important formulas or theorems that are simpler in their preferred notation. But lacking any explicit idea that mathematical notation needs to be optimized for the limited short-term memory capacity of human brains, they spend what I think is too little time on such “global” arguments and way too much on “local” ones – that is, whether the ratio that π or τ expresses is “more fundamental”.
I think this local argument really rests on a sort of lurking Platonism, a belief that a mathematical formula is a kind of statement or claim about ideal forms at least some of which have an existence independent of the formulas. Nobody in the dispute can quite bring themselves to utter the claim that π is “real”, whereas 2π is just a derivative arrangement of symbols; nor does one hear the opposite claim that τ is “real” but τ/2 is not. But that sort of essentialism is stooging around underneath the arguments the disputants do make, denying its own presence but nearly impossible to miss.
If the last century and a half of mathematics has taught us anything, though, it’s that Platonism doesn’t work. Kurt Gödel put the final bullet through its head with his Incompleteness Theorem in the 1930s, but it had been living on borrowed time ever since Bertrand Russell blew up Frege’s axiomatization of number with a simple paradox in 1902. Mathematical Platonism has since almost disappeared as a philosophical position, but not as a psychological one; I’ve noted before that mathematicians then to be formalists in theory but Platonists in practice. In disputes like τ-vs.-π the tension between these positions surfaces, because arguments about the notation of mathematics have a natural tendency to slide over into arguments about its ontology.
Having restated the underlying problem in a way that I hope clarifies the dispute, I will now take a position on the merits. I think the Tauists have the better of the argument. I don’t think I’m being influenced too much by the fact that their side gets to make clever puns about Taoism, difficult though that lure seems to be for anyone involved to resist. It really does appear to me that the τ notation yields a net simplification.
Much more importantly, though, I think the best way to resolve this dispute is to throw out all the essentialism and the arguments about what π and τ “really mean” geometrically. We should focus ruthlessly on the global question: what notation makes our formulas simpler?
That way of thinking about the problem implies an answer to the question about what we mean when we say “π is an important mathematical constant”. We mean that it shows up repeatedly in simple formulas – and that replacing it with an equivalent expression that is not one symbol (such as, say τ/2) would involve a loss in concision with no gain in expressiveness.
The Tauists claim that changing to τ would actually gain some concision. Very well then; let’s do a systematic audit. Representatives of the Tauists and the pi partisans should be locked in a mathematics library until they choose a list of books and papers that covers trigonometry, calculus, and analysis. Then, the burden should be on the Tauists to translate the entire pile into τ notation. Then, both sides should count symbols, checking each others’ work. Most compact notation wins!
Or, to put it a different way, if you’re going to get involved in the τ-vs.-π dispute, beware of circular arguments.
> if you’re going to get involved in the τ-vs.-π dispute, beware of circular arguments.
I really should have seen this coming… and one might suspect your motives for this entire post.
I did not in fact write the entire post to set up that pun, but I admit that your suspicion was a priori quite reasonable. :-)
The obvious theme song for the pi party by Kate Bush.
Nice overview… I’d seen this argument going on for a while. Amusing, but useless, in my opinion. You want to use ? a lot?
Just write
#define ? 2*?
In your work and use it when necessary. No hassle needed. If it’s a good idea, others will use it.
On the other hand, typo: 4th paragraph, 2nd line :: “math ematical”
Also, although I’m not a native English speaker, I think the phrase “until they choose a list of books and papers that covers trigonometry, calculus, and analysis” is a bit clumsy. Maybe “covering”, instead of “covers”? I just tripped upon that sentence in my head.
Eric, they’re arguing about circles, not in them.
The symbolic weight methodology, to me, tends to favor τ for a different reason.
In my line of work, I have to convince people to do arithmetic operations as part of the entry price to have fun. I am probably more conscious of the symbolic weight of simple mathematical statements than most people on the planet.
Counting is easier than addition.
Addition is easier than subtraction.
Subtraction can be done, but combining addition and subtraction into the same game mechanic will slow down play considerably. (“I have +2 for flanking, and -2 for lighting…”) is surprisingly fiddly in play.
BIG GAP HERE
Subtraction and addition in the same operation is easier than most forms of multiplication. For example, it’s better to express a critical hit as “+100% damage” rather than “x2” damage. Not only does it avoid edge cases (where you have two critical hit multipliers that both apply) it’s faster at the gaming table because it’s addition.
Multiplication by 10, 5, 4 and 2 are light weight enough to be usable. If you’re using a multiplicative operator other than one of those four digits, you need to change something earlier in the process. When in doubt, make it addition.
BIGGER GAP HERE
Division is only tolerable when A) you’re dividing an integer and expect an integer outcome, or are rounding to an integer outcome, and B) the divisor is 2 or 10. I specifically built some parts of my games for modulus division to avoid these problem.
GINORMOUS GAP HERE
Square roots, cube roots and exponentiation are probably not game friendly to most people.
When in doubt, transform your equations to use one of the first two items on this list.
By that logic, multiplying τ by 2 is much less annoying that dividing π by 2.
Since I’ve been aware of Tau, I’ve been an advocate.
I like the fact (promulgated at tauday.com, IIRC) that you can gently introduce Tau into your math, with one of those up-front precondition statements:
Let Tau = 2*Pi
Then you use Tau everywhere, and (at least in some of the work I do, there are so many missing 2 * that it’s not even funny. If everybody does this one simple thing, it becomes a self-propagating meme, until eventually somebody is going to put tau=2*pi in (for example) the python standard math.py.
You have taken the theory side. Ken Burns has taken the general public side. Let me take the practical (engineering) side.
Whether Tau is more fundamental than pi or not, it’s an incredibly useful abstraction. I deal with periodic data, and much of the math revolves around what happens in one period. Not half of one period.
In closing, let me present some quotes from the most important practically unknown electrical engineer ever.
Oliver Heaviside was one of those geniuses who had an intuitive understanding that transcends that of any mere mortal. It was focused, laser-like, on understanding electricity. He used math as a tool in this and, in true open source fashion, published what he was doing and his results.
He was always at loggerheads with the mathematicians, who didn’t figure out until much later that there actually were good theoretical underpinnings for Heaviside’s work.
I don’t operate at nearly the level as a Heaviside. Nonetheless, I share his view that math should be a practical tool for getting things done, and even if there are sound theoretical arguments about why pi is better than tau, I don’t care if tau is better for actually getting things done.
Hmm, that last paragraph didn’t read quite right. Let me try again:
I don’t operate at nearly the level as a Heaviside. Nonetheless, I share his view that math should be a practical tool for getting things done, and even if there are sound theoretical arguments about why pi is better than tau, I don’t care, and I will continue to use tau if it is better for actually getting things done.
Godel showed limitations of formal systems. What does it have to do with Platonism? That you’re unable to grasp the truth in your formal system does not imply it doesn’t exist.
As an academic I had the instinctive, gut-level reaction “How much free time do these people have?”
They need to get their asses on a couple of faculty committees and start doing their share of the scutwork.
Jeez.
PS: The Tau Manifesto link doesn’t work. I think you need to add an “m” so it’s .com?
*fwap*
Give the divisions already present in the world, isn’t abetting the aims of the Second Revolutionary Tauist Jihad a little….irrational?
I think e-to-the-i-pi equals -1 identity the deal all by itself. e-to-the-i-tau equals 1, but that has significantly less mysterious beauty.
Also: when physicists got tired of writing h over 2pi all the time, they just started using h-bar instead.
@Paul Sand:
It is “e^{ i \pi } + 1 = 0”, and expression that includes all important constants (for certain values of “all” ;-))
@ Paul Sand:
@ Jakub Narebski :
> It is “e^{ i \pi } + 1 = 0?, …
I think it is i*pi, but I was going to bring it up.
Lots of physics use angular velocity in radians so it is f/tau.
Maybe we should compromise, and define sig-rho as 1.5*Pi or Tau*3/4.
@Paul, Jakub: Also, it includes the three fundamental binary operations: addition, multiplication, and exponentiation.
I think it was Hardy who stressed this aspect of the beauty of this formula of Euler.
@Ken, I prefer the phrase International Neomathematical Tauist Jihad, or INTJ. Damn those INTJs…
@Paul Sand: e^(i*tau) – 1 = e^(i*2*pi) – 1 = (e^(i*pi))^2 – 1 = 1 – 1 = 0. In other words, Euler’s famous equation looks equally beautiful if you simply substitute tau for pi, and I’m sure the Tauists (do they refer to the other side as Confusionists?) have pointed this out. The proof of the tau version might be slightly more muddled, however.
One aspect I haven’t seen addressed is legacy support. Setting aside whether introducing division imposes a higher mental load (e.g., I seem to recall integrating over fractions of pi more often than whole pis anyway), does any benefit of more compact representation outweigh the overhead of switching? This seems to fall into the same category as whether programming keywords should be in English or Esperanto.
It’s a pity the symbol tau wasn’t used from the beginning instead of pi, because then we could now introduce the novel concept that pi=2*tau, with the lovely mnemonic that a pi symbol looks like two taus glued together (like a “w” (double-u) is actually a double-v). Reminds me of one of my favorite words, “nybble”, which if you have never run into it before, is a 4-bit word. I.e. a nybble is “half” of a byte.
RubeRad: Except that pi=tau/2.
Now, having read both manifestos, I’ll add this: This is exactly analogous to the constant carping from Europeans that we in the US should adopt A4 paper as our standard instead of the 8-1/2×11 inch size. They cite as primary justification for this the fact that the A-series paper sizes are derived by cutting each one in half to reach the next smaller size. Whoopee ding. The benefits of such a conversion are vastly outweighed by the immense costs: let’s start by replacing every filing cabinet in existence.
The same goes here. What are the costs of conversion, and what are the benefits?
I would argue that Eric’s analysis is an important step, but only a first step. Once that’s done, the benefits need to be weighed against the costs, and only if the benefits substantially outweigh the costs is the conversion warranted.
Beautiful article and post.
I guess I was a tauist as a kid. I remember asking the math teacher why all the circle/trig equations were not expressed in terms of 2*pi rather than in terms of r (radius) and D (diameter) and heard the familiar propaganda: How pi was the most important constant in mathematics ever etc.
Tau though is heavily used by electrical engineers to signify time constants. It is even more proliferated that its use as torque by mechanical engineers. We need to coin some new symbol. If enough artists go at it (like they did for the euro currency) they will come up with the perfect symbol for tau.
I take it seriously when ‘contravariant’ tensors are properly renamed to ‘covariant’. First things first.
@Paul Brinkley: But then we wouldn’t get the puns off of second (tau is pi * 2), revolutionary (it’s about the path defined by a curve revolving around a fixed point) and so on.
I think that one sentence had the highest pun density of anything I’ve written in years… :)
I can never decide whether I love or hate Heaviside.
as an APL(ish) programmer, i endorse this definition of merit
@RubeRad: and a quartyr is half a nybble (two bits)
Perhaps the new symbol (instead of tau) could be called Ti (pronounced “tie”) where Ti = 2*Pi.
The only thing needed is is a new math character to represent it.
HTML hint for people who have trouble making these symbols:
We are talking about these:
π=π
τ=τ
We are NOT talking about these:
Π=&Pi
Τ=Τ
@Jay Maynard:
“They cite as primary justification for this the fact that the A-series paper sizes are derived by cutting each one in half to reach the next smaller size.”
Well, that’s true of US standard paper sizes as well. Cut 11×17 in half and you have 8.5×11. The difference is that the A-series paper sizes are all in the same proportions, and US paper sizes aren’t.
“One aspect I haven’t seen addressed is legacy support.”
Absolutely. The folded scales on my K&E Log-Log-Duplex-Vector slide rule are offset by pi. Shifting them by tau would render them much less effective. Also, my HP-11 calculator has a pi function, tau would require some mental translation when entering formulas….
Converting from one system to another can be done and often has merit. There are many countries that have gone from Imperial to Metric. Even the UK is getting there, inch by inch.
Italy is gradually changing its electric plugs to be compatible with the majority of European countries. Their new electric outlets accept both old and new plugs. Old plugs are not sold any more. If you need to plug a new device into an old socket, you need to use an adapter.
Sweden changed from driving on the left to driving on the right in 1967. For some reason we all had left hand drive cars and drove on the left. The switch was a huge improvement (though some big crossroads for traffic still feel funny because they were built for traffic going the other way).
Changing to the ISO paper sizes would be a fairly simple thing, since you could do it gradually. It is not as if the investment in a double set of hole punches is huge. Filing cabinets usually hold either size and the European style binders are superior to anything I have seen in the US. A change would actually pay off in many small details. One of them is that you can actually calculate paper weight and paper area much more easily. For instance, normal printer paper weighs 80g/m2. An A4 sheet is 1/16 of a square meter (A0 is defined as being 1 m2, A1 is haft of that etc), so weighs 5 g. I know I can fit 3 of these in an envelope and still be under 20g, which is the lowest postage limit. I can work things out in my head from just a couple of basic definitions and some simple rules.
Fortunately the US will never adopt the ISO paper size system, so we Europeans get to keep one of the few competitive advantages we have.
I’m not sure that simple symbol counting is the right metric. Clarity counts, and sometimes more symbols makes for greater clarity. My favorite example of fewer symbols and less clarity is the use of juxtaposition to denote multiplication. As Dijkstra observes it makes teaching the difference between:
3½ and 3y and 32
somewhat harder. Not to mention the problems if you want to use multi-letter symbols and traditional function notation.
For people interested in notational issues I do recommend the last revision of Dijkstra’s memo on notation: http://www.cs.utexas.edu/users/EWD/ewd13xx/EWD1300.PDF . I don’t often agree with Dijkstra, but I think he hits it pretty much bang on in this memo.
(This is not your fault, I’m sure, but it was really distracting here the way the pi glyph (in FF5 on Win7) looks a lot like “n”, rather than the more “traditional curly-pi” glyph.)
why not the more descriptive ‘rev’ or revolution?
also, why do mathematicians avoid descriptive names for variables? to accomodate that, you’d have to count the nodes on the syntax tree rather than the symbols.
what happened to my first post?
I have to side with the Pi-ists. This shirt is why.
@Ken: I’ve also been aware of the issues with doing arithmetic in gaming for some years now, with a usenet post by John Morrow crystallizing it for me back in 1995. I mostly agree with your evaluations, but I’d say that multiplication by a single digit is only slightly worse than subtraction, to the point where having to deal with larger numbers can overcome the advantage of subtraction or even addition.
So I’d say that multiplying by 2 is about the same as adding 100% in terms of difficulty “in the heat of play” (sometimes a bit worse but sometimes a bit better) and that multiplying by 3, 4, 5, etc. is definitely better than adding 200%, 300%, 400%, etc. Especially since the latter includes a hidden multiplication anyway.
But I very much agree with the “BIGGER GAP HERE” between multiplication and division.
Meta comment: I’m noticing an elevated level of Typos in both the Google+ posting and this one. Have you switched input technologies recently?
My typo percentage is up slightly because I am on the road and blogging from my laptop and smartphone, not my comfortable workstation. The slight degradation in the ergonomics and visibility of text makes a difference.
That you only recently became aware of this argument speaks volumes.
You might be interested in Wildberger’s “rational trigonometry”, in which he eschews the concepts of angle and distance for spread (square of sine of angle) and quadrance (square of distance). It gets rid of all the sines and cosines! Woo-hoo!
Of course that proposal is quite a bit more radical than this tau-vs-pi thing.
m, yeah, but while τ might simplify some maths, it unnecessarily complicates others, right?
For example, the volume of a cylinder is conventionally π*r^2*h. How is (τ/2)*r^2*h any better? Or even the proposed (1/2) * τ * τ^2 * h?
Am I missing something important?
I’m not sure it makes any real, qualitative difference at all, to be honest. Other than some interesting humor value.
Note that IANAM.
In fact, wouldn’t computing it the τ way be more computationally intensive with current coding techniques?
> In fact, wouldn’t computing it the ? way be more computationally intensive with current coding techniques?
It’d be trivial if the language or library supported it directly; we’re talking division and multiplication by two here.
In the applied world, it is a bit odd to use half a revolution…..OTOH:
piR^2
vs
1/2tauR^2
<elegant.
1/4piD^2
vs
1/2tauD^2
Perhaps in modal analysis there may be some advantage in the notation change, but you still have harmonics…divided by two.
I'm a close enough for practical purposes type….discretization rules…everything else is just mental masturbation.
@esr:
What exactly does this mean? Fewer distinct symbols or fewer total symbols?
I assume you mean the latter (fewer total symbols), in which case a strategy which beats both the &pi-strategy and the &tau-strategy is to use both &pi and &tau depending on context. But in that case you can just keep introducing new symbols, and at some point you lose the ability to see the relationships between formulae because they have been hidden by the notation.
If you meant the former (fewer distinct symbols) then you can replace &pi and &tau with Madhava-Gregory-Leibniz series, which can’t possibly be the Right Thing To Do.
>What exactly does this mean? Fewer distinct symbols or fewer total symbols?
Fewer total symbols. The use-both strategy is only a clear win if there is no cognitive cost to increasing the number of primitive constants (such as π and e) that you use in your notation. In fact I think that cost is quite high.
Crap. I thought cut ‘n paste would work on the unicode.
@Morgan:
From an engineering perspective, if you’re doing a lot of area calculations, sure — keep using π if you find that simpler. (Your cylinder volume example is trivially an area calculation.)
Perhaps not. But if you read and thought about the area calculation section at tauday.com, you might concur that, from a pedagogical perspective, it’s easier to teach how we get to the area with τ.
And it may be that, if students grow up with a better fundamental understanding because of better notation, they will make even more discoveries. This obviously isn’t as big a deal as switching from Roman to Arabic numerals, but it might be useful.
In my world, there are two types of calculation. Infrequent and DSP. But area calculations are typically infrequent, while most DSP calculations involve 2*π somewhere. Typically, however, this is already maintained as a separate variable, either manually by the programmer, or automatically by compiler loop optimizations. So I don’t think machine efficiency is a good argument for or against.
The only real arguments for or against change boil down to (1) pedagogical, (2) communication with other humans (not machines), and (3) communication with yourself, e.g. when you are designing something. π currently wins (2) for trivial, infrequent communications, because for most people it currently wins (3) hands-down. The tauday people make a good case that τ wins (1), and for a lot of what I do, it now wins (3), and can win (2) if I’m sharing DSP-heavy thoughts with a co-worker and start off with “let τ = 2*π”.
There are other arguments I discount, which mostly seem to boil down to the Sherlock Holmes admonition not to fill your head up with useless stuff.
@TMR:
The elegance varies according to use-case. In any case, I think you meant 1/8 τ d^2.
Which brings up the point that if you’re going to persist in using area as your benchmark, the real constant you want is probably π/4 (or τ/8) for two reasons: (1) most people think of circles as having less area than the corresponding square; partly because (2) in real life it’s damned difficult to measure the radius of some random device or piece of ground. People measure the diameter and divide by two.
@ Patrick:
The elegance varies according to use-case. In any case, I think you meant 1/8 ? d^2.
Yes. You are correct…should not watch a movie and type at the same time.
Which brings up the point that if you’re going to persist in using area as your benchmark, the real constant you want is probably ?/4 (or ?/8) for two reasons: (1) most people think of circles as having less area than the corresponding square; partly because (2) in real life it’s damned difficult to measure the radius of some random device or piece of ground. People measure the diameter and divide by two.
Hydraulic diameter is a useful concept here.
Planimeters are also very intriguing in this instance.
>The use-both strategy is only a clear win if there is no cognitive cost to increasing the number of primitive constants (such as ? and e) that you use in your notation. In fact I think that cost is quite high.
Exactly, any metric needs to include both total symbols and number of distinct symbols. Although if you’re only interested in comparing π and τ the number of distinct symbols is the same in both cases.
Another thing to take into account is the number of patterns. The tauists argue that the formula for the area of a circle is simpler with τ, despite containing more symbols, since it follows the same pattern as the formulae for kinetic energy, Hooke’s law etc.
>The tauists argue that the formula for the area of a circle is simpler with τ, despite containing more symbols, since it follows the same pattern as the formulae for kinetic energy, Hooke’s law etc.
Right, this means that in principle one could create a spanning notation for the whole family that would lower the overall Kolmgorov complexity of the expression of the list of theorems in the mathematics they span. So this actually gets us back to lowest total symbol count.
>Right, this means that in principle one could create a spanning notation for the whole family that would lower the overall Kolmgorov complexity of the expression of the list of theorems in the mathematics they span. So this actually gets us back to lowest total symbol count.
Nope, you have to add a new symbol for the spanning notation. You’ve reduced total symbols but increased distinct symbols. In this case I think the trade-off is a win, but in order for a metric to reflect that it must include both concepts.
>Nope, you have to add a new symbol for the spanning notation. You’ve reduced total symbols but increased distinct symbols. In this case I think the trade-off is a win, but in order for a metric to reflect that it must include both concepts.
I don’t think the concept of “new symbol” you’re assuming is as yet well enough defined for you to assert that. Do you intend a “symbol” to be just a text macro on the space of WFFs in the notation or something else?
I’m no mathematician but to me the symbol with the easiest pronunciation wins. Pie. Simple.
I should have said simple as pie. :)
I guess this might be naive, but I thought that, as a rule, one should factor anything one can factor. You can’t factor pi, but tau can be factored as 2*pi. So pi is more fundamental, in the sense of simplifying the expression.
>I don’t think the concept of “new symbol” you’re assuming is as yet well enough defined for you to assert that. Do you intend a “symbol” to be just a text macro on the space of WFFs in the notation or something else?
I think I see what you mean now: if you create a new notation then the symbol count already includes the symbols required to define the notation. So there’s no need to account for the new notation separately.
Earlier you said:
>The use-both strategy is only a clear win if there is no cognitive cost to increasing the number of primitive constants (such as ? and e) that you use in your notation. In fact I think that cost is quite high.
The additional symbol cost of the ‘use both’ strategy, relative to either the ‘all π’ or ‘all τ’ strategies, is symbol_count(τ = 2π). But presumably there are more than that many places in mathematics where 2π could be shortened to τ, or where τ/2 can be shortened to π.
Therefore the ‘use both’ strategy has a lower symbol count than either ‘all π’ or ‘all τ’.
We both agree that the ‘use both’ strategy is probably inferior because of the increased cognitive costs. But these cognitive costs don’t seem to be reflected in the symbol count.
+Patrick Maupin:
I only marginally agree with the tauday area argument, mostly because τ brings the area calculation in line with other quadratic forms. But only marginally. There is something to do be said for tradition, IMHO. For example, why do programs that fail to find a necessary return result code of 2 on abort? Tradition. There is no reason why some other result code couldn’t be returned why there has to be a result code at all: the program could conceivably communicate that condition with the calling process via some other mechanism.
I had completely forgotten about DSP calculations. You’re right: most of those involve 2π and conceivably the language/library could provide a tau() function instead or in addition to pi()
>The notation with the lower symbol count wins,
I’m not a mathematician, but I must say I wonder about this conclusion. In many other fields semantic density is not an unadulterated good. Consider for example:
for(int i=0,s=””;i<0;++i) s+=(i%2==0?"Fizz":"Buzz");
This is pretty dense, but also pretty hard to understand. Spacing it out makes it much easier to understand. (I'd offer a spaced out counter example here, but the likelihood is that wordpress would chew it up, and mock me with the irony.)
It is often true that high density in English can also make for wearisome sentences, though for sure, wordiness also has that effect.
It is my experience that a little extraneous redundancy in notations can allow the sentence "a little breathing room" as a commentator on a different blog once said. A few extra symbols (such as white space or comments in code for example) allow the mind some rest time between the parsing and semantic analysis phase, and that is often a good thing.
I'm sure we have all had the experience of reading over a sentence a dozen times trying to parse out the exact meaning of the words, their semantics and referrants. Sometimes a few extra words leads to clarity, even at the expense of concision.
But, I'm not a mathematician, so perhaps things are different there.
BTW, my code was wrong it should be:
for(int i=0,s=””;i<10;++i) s+=(i%2==0?"Fizz":"Buzz");
I guess I mocked myself without wordpress' help :-)
@Sam:
I gave an example earlier. Let me clarify it. Say I want to know the cross-sectional area of a pipe. First I’m going to find the diameter. Then I’m going to square this. If the pipe is square I’m done, but if the pipe is circular, then I need to “take some off” of this final answer to compensate. In order to do so, I’m going to multiply my answer by π / 4.
So according to your logic, the “fundamental” constant should apparently be π / 4 simply because it’s a smaller number.
But, as I mentioned earlier, if, for example, you want to know the cross-sectional area of
@pete:
> The tauists argue that the formula for the area of a circle is simpler with τ, despite containing more symbols…
See, I don’t think it does contain more symbols in real life.
I’ve mentioned this multiple times, but seriously, in real life, when is the last time you calculated the area of a circle? I think that for me, the last time would have been about water storage in cisterns. Let’s see, if I have a 6 foot high 6 foot diameter cistern, then the volume is… (divide 6 foot diameter by two to get the radius, then square it, etc.).
In general, the diameter is a much more useful metric for the cistern than the radius. Want to know whether it will fit where you are planning on placing it? Want to measure another cistern to see if it is bigger or smaller? (OK, in that instance, you might use the circumference.)
The diameter, like the circumference, is directly observable. In fact, this fact is the real reason why π is considered to be fundamental — it links the easily measured diameter to the equally easily measured circumference. But, unlike the diameter, the radius depends on this imaginary non-delineated point at exactly the right location inside the cistern.
So I don’t agree that somehow π * r^2 shows that pi is better than tau, because the starting point for pi, and for the how people interact with real-world circular objects is the diameter, not the radius.
So the realistic formulation for the area of a circle is π / 4 * (d ^ 2).
Now, when dealing with more advanced math, it is often useful to use the radius rather than the diameter. But if you are using the radius then the fundamental linkage of the radius to the circumference is tau, not pi. And the new formulation for the area, τ / 2 * (r ^ 2), is no more complex.
a slight issue is that tau as a number can clash with the many other instances in which that letter is used.
there are only 24 letters in the Greek alphabet and 26 in the roman. by convention many of these have meanings which are specialised (f, g for functions; x, y, z for spatial coordinates etc.)
i suggest ‘tr’, ‘turn’ or ‘rev’.
also to those who object about the formula for the area of a circle:
the area of a sector is 1/2 * theta * r^2 (a result that is shared with triangles of ‘infinitesimal’ angle which are used to perform polar integration because dA=1/2*sin(theta)*r*(r+dr) with sin(theta)->theta and dr->0 as theta->0), and the circumference is theta * r. 1/2 rev r^2 is then just a result of a more general formula.
The principle is called the mininum message length criterium and it is mathematically sound. The relation with Kolmogorov complexity is obvious.
https://secure.wikimedia.org/wikipedia/en/wiki/Minimum_message_length
It does account for the number of symbols.
>The principle is called the mininum message length criterium and it is mathematically soun. The relation with Kolmogorov complexity is obvious.
I was, actually, not aware of the work on axiomatizing MML – although it was published before my time, metamathematicians in the 1970s didn’t talk to computational-complexity theorists much or vice-versa. Actually the two groups still don’t communicate as well as they should.
But yes, MML is what I was driving at.
@Jessica:
You make a good point that terse prose can be difficult to decipher, but I think this is somewhat orthogonal to what is being discussed.
Terse or verbose prose describing the same concept is merely a matter of how the concept is encoded in the communication channel.
There is no question that redundancy, whitespace, etc., can be huge aids to communication. This is true both with human communication and machine-to-machine communication, where the technique is known as “forward error correction.”
But that’s redundancy in the communication channel. I don’t know of any basis for believing that redundancy is inherently useful inside of a concept. In fact, I think most here would agree that the most valuable concepts are those that are applicable to a wide range of fields, thus reducing the total number of concepts you need to know. That’s part of why stories like this one are so funny — it wouldn’t be funny if “medical calculus” was a necessarily distinct concept from “electrical engineering calculus.”
As esr points out informally (and Winter gave the formal name for), the “best fit” for a view of a concept is likely to be the one with the smallest required message length. But that doesn’t mean that redundancy isn’t useful when we are explaining or discussing the concept.
My God, that calulus thing is so funny yet so wince-inducing. Imagine how many emails that poor person got saying “Dude, open a first-semester Calc book.” Ouch.
It also doesn’t inspire much confidence in the journal’s review process!
+Patrick Maupin
In large part because what I have to do is generate search areas, and my detection parameter is a 1/r^2 or 1/r^4 parameter emanating from a fixed point, for me, pi is more useful, because I’m already starting from a known point of origin.
Same thing applies when calculating orbital velocities. You know what your centroid point is, sometimes you don’t know what your circumferential orbital distance is (and you have less use for that than you do orbital velocity and angular momentum, in any case…)
@Ken Burnside:
Isn’t (2 pi) ** 2 used in the standard gravitational parameter?
>the “best fit” for a view of a concept is likely to be the one with the smallest required message length.
Right but the specific question I was addressing was the claim Eric made that the notation with the highest semantic density was the best notation, in the context of communication to people. (He made mention of brains unpacking the concept for example.)
I think that is a very strong claim, and I doubt it is unreservedly true.
@Jessica:
I would agree that his claim, filtered through your interpretation, certainly isn’t true. However, his claim, filtered through my interpretation, is true. (Whether the claim itself as presented and clarified through subsequent argument is true or not is something I can’t decide without re-reading a lot of stuff I can’t be bothered with at the moment.)
The thing is that you need to find a sweet spot between the size of the global symbol set and the number of symbols required by the average message using those symbols. If applied properly, I think the minimum message length criterion comes very close to fulfilling this function. For example, English can describe any word with 26*2 symbols (sometimes requiring context disambiguation), while traditional Chinese writing seems to require somewhere around 3500 symbols (probably still requiring some amount of context disambiguation).
I think that the alphabet is a major step forward. Arabic numerals are a numeric analogue of the word alphabet, allowing a number to be built up from a small set of symbols, where the symbol means one thing, and the place of the symbol within the word means something else.
I found an interesting (if long) paper on this topic called The Representation of Numbers when I was researching arguments for which radices to support for Python PEP 3127: Integer Literal Support and Syntax.
“””
Converting from one system to another can be done and often has merit. There are many countries that have gone from Imperial to Metric. Even the UK is getting there, inch by inch.
“””
F’ing BRILLIANT.
Next time I’m in a pub I’ll raise a 500ml in your honor.
Then you have this:
http://www.maa.org/devlin/devlin_06_11.html
@Oblivious:
> http://www.maa.org/devlin/devlin_06_11.html
Interesting theses there. A bit off-putting that he’ll share all the proof with you if you just buy his book.
And then, of course, there’s this:
I’d have to see a cite for this. I certainly believe that the average adult realizes (correctly, btw) that, in the natural number domain, you can achieve the same results as multiplication with repeated addition, but I also believe that the average adult would not even consider the use of repeated addition to multiply two four-digit numbers together (at least, not without interspersing shift operations).
He expands on this theory a bit in other column. For example:
Frankly, I find this stupid. Next thing you know we’ll stop teaching classical mechanics, because it doesn’t work at the sub-atomic level. Note also that he doesn’t have a better idea — he freely admits that he doesn’t know the right way to teach multiplication, but claims that teaching repeated addition is wrong because it has to be “fixed” later.
But why reserve all the vitriol for the teaching of multiplication? The kids who go on to learn abstract algebra will learn, not only that “*” doesn’t necessarily mean what it used to, but neither does “+”. Oh, and by the way, not only does “-” not work the same any more, you can’t even use it at all in this context because the results are undefined. It’s worse than finding out there’s no Santa Claus!
One more quote from the original article you referenced:
OK, which is it? Initial explanations of multiplication as repeated addition scar people for life, or ordinary working adults have a 98% accuracy rate with numerical calculations?
Another book plug. My theory is that people can’t do math symbolically because they tried to learn it from a git like him, and the first thing he told them is that you can’t multiply via repeated addition. Cue Barbie moaning “Math is hard!”
I was curious about the luddites who use “repeated addition” in their teaching of multiplication. Devlin had mentioned that several of them noticed and commented on his previous articles about the subject. I found a good post (with great comments) here:
http://letsplaymath.net/2008/07/01/if-it-aint-repeated-addition/
One of the comments led me here, where we find out why, in our gut, we know that Devlin’s approach might have issues:
http://homeschoolmath.blogspot.com/2008/07/isnt-multiplication-repeated-addition.html
Devlin says we should teach multiplication as scaling. Fair enough. But as this lady asks, if I’m teaching small children by using concrete objects, and I multiply a “car” by two, do I get: a car that’s twice as long? a car that’s twice as tall? a car that’s both twice as long and twice as tall (that an adult would say was four times as big)? Or simply the normal answer of “two cars”?
@Patrick Maupin:
Obligatory xkcd. Math is pretty universal, of course.
Which touches on my personal pet theory that if more people learned some basic computer science at an earlier age they would have less problems doing math symbolically. (This is the part where people usually look at me like I’m crazy)
> This is the part where people usually look at me like I’m crazy.
Do programmers look at you like you’re crazy? Or just “normals.”
I’ll go you one further. If you hand somebody a box with a Z-80 and a front panel with switches and blinky lights (well, OK, that may not be practical — maybe just an ancient programmable calculator like a TI-59) and have fun with it with them together, they will develop an intuitive understanding of the utility of symbols before they even know of their existence: “Oh, you mean I can name this memory location?”
@patrick maupin
>I multiply a “car” by two
It’s obvious. You get a car that’s twice as fast or twice as cool.
“…but I also believe that the average adult would not even consider the use of repeated addition to multiply two four-digit numbers together (at least, not without interspersing shift operations).”
Yes, that’s exactly what the average adult used to do, in the days of mechanical desk calculators (at least the reasonably small ones). They had a repeat key that was designed for that. (You used the zero key to perform the shifts.)
Of course, there were monster ‘desk calculators’ (about 1 foot by 2 feet by 1.5 feet high) that had a multiplication key, so you could multiply 2 times 3 in about 10 seconds and get 5.9999999. (Division took a lot longer.)
@Jim Hurlburt:
> It’s obvious. You get a car that’s twice as fast or twice as cool.
But a car that’s twice as fast is automagically at least 1.2 times as cool, which must mean that it’s 2.4 times as much car :-)
Unless, of course, the slower car is a Model-T. Hmm, going to need a bigger equation.
@LS:
>> at least, not without interspersing shift operations
> Yes, that’s exactly what the average adult used to do…
And it’s still what the average 8-bit firmware programmer does. The average 32-bit desktop programmer simply uses hardware to do this for him (analogous to the bigger desk calculators you mention), but you can bet that the hardware designer, whether building a calculator or building a chip, thought in terms of shifts and adds, even if the shifts are performed in space (additional circuitry) rather than time (additional clock cycles).
And that’s true even for the hardware that operates on floating point numbers. Sure, there are other implementations (log tables, analog computers that actually use logs and anti-logs, etc.) but those aren’t mainstream.
This is seriously off topic, so feel free to delete this comment, but I wanted to bring this to your attention:
The ITC has recently found that Android is in violation of two patents held by
by Apple. A good analysis here:
http://fosspatents.blogspot.com/2011/07/itc-judge-finds-htc-in-infringement-of.html
http://fosspatents.blogspot.com/2011/07/these-tables-show-how-android-infringes.html
One of these patents, though puts me in mind of a common feature in many
text editors, such as GNU Emacs and Gedit, as well as other software
like email clients and IRC clients. These programs automatically detect
Web URLs embedded in text streams, highlight these URLs, and make them
“actionable,” which would appear to infringe on the patent.
The patent was filed in 1996. It could be that code to recognize certain
data structures in a text stream, highlight them, and make them
actionable, has been in GNU Emacs since before that year, thus
constituting prior art. I really don’t know when that feature was added,
but it’s worth looking into.
ESR, since you used to be in on the Emacs modding scene, would you
happen to know offhand when this feature was added, or would you know
who to ask or how to find out?
I’ve also sent an inquiry about this to the FSF.
>ESR, since you used to be in on the Emacs modding scene, would you happen to know offhand when this feature was added, or would you know who to ask or how to find out?
I don’t know when it was added, but I can tell you how to find out. Search for “URL” in the Emacs NEWS and OLDNEWS files.
Patrick: The more interesting point from the homeschoolmath site is the comment from Joe Niederberger, which is that if you really dig into number theory, it turns out that basically, yes uh-huh multiplication starts out as repeated addition. Starting with the natural numbers, the way you multiply x * y is to write the moral equivalent of x + (x * y’) where y’ is the previous y minus 1, and then recursively expand the remaining multiplication until y’ is zero. Multiplication is then expanded into further domains. But the very most basic form of multiplication there is at the most basic level of math is indeed repeated addition. And is that not exactly what we are talking about teaching?
I see no reason not to essentially teach children the basic truth; here’s repeated addition, here’s how we can extend it as we add more types of numbers. Of course, this would require that the people who are teaching math actually understand math first, rather than just having it mechanically drilled into them by their teachers, which as we all know is just too much to ask of a teacher.
@Jeremy Bowers:
I agree absolutely. Joe Niederberger made a lot of good points in the comments on the letsplaymath site, and Maria Miller copied just one of his cogent comments onto her homeschoolmath site. (I found the letsplaymath site first, but Maria Miller’s post, including the Joe Niederberger comment, was so compelling, I had to add a link to it.)
Many of the educators commenting on the letsplaymath site (but particularly Joe Niederberger) also seemed quite knowledgeable about multi-modal teaching about multiplication (which you would expect, given that they were engaged enough to discuss their work on the website), and gave several lucid arguments about why repeated addition is a valuable part of the arsenal, both from the education perspective and from the math perspective.
But if you go back and re-read Devlin, he had apparently already read the comments on both sites and was highly dismissive of all of them, which is incredibly counterproductive.
I specifically pointed out Maria Miller’s example of multiplying cars because I think it is brilliantly targeted at people who don’t have either an education background or a math background, and who see the education specialists and the math specialist arguing over really esoteric stuff, and can’t decide who to trust. I think that example puts things in perspective in a way that any layman can understand, and could be a huge aid for smart but undereducated people in the determination of who is bullshitting them.
When a non-specialist sees the NPR math guy still saying
(thus implying that the only comments anybody ever made on the issue either lauded him or were made by complete cranks)
and simultaneously sees Maria Miller saying:
and then describing those problems in detail, I think it should be pretty obvious to most people which of those two you would be better off entrusting your children’s education to. (And a few years with Maria Miller followed by a few more with Joe Niederberger would probably give your children a much better mathematical grounding than most Americans have.)
@Jeremy:
I made a longer reply to your comment, but it seems to have been eaten by wordpress. Just as well — if it’s one of those comments that shows up ten hours later, I’ll realize that this later synopsis contains everything worthwhile in there anyway…
Short version:
I agree completely that Joe Niederberger’s comment gives a rational explanation for the use of repeated addition. Several of his other comments at the letsplaymath link were on-point as well (especially the comments about how Devlin’s own “scaling” approach might not work all that great once the domain is extended to complex numbers), but I thought Maria Miller’s post was fantastic, because it is perfectly understandable by a completely non-technical audience. In other words, the people who might succumb to an appeal to authority now have something understandable to fall back on to back up their gut feelings.
That’s perfect. I found this in NEWS.23.1:
*** goto-address.el provides two new minor modes, goto-address-mode and
goto-address-prog-mode, which buttonize URLS and email addresses.
Annoyingly, that only goes back to 2009. I found references in earlier changelog files to a feature that allows you to load the current word as a URL, but that requires you to explicitly tell Emacs to treat it as a URL, so I don’t think that’s prior art.
I guess I’ll look in other software.
[1]http://www.gnu.org/software/emacs/NEWS.23.1
@Max E:
Are you just a concerned citizen or are you working with one of the parties? After looking at it, I may actually have coded some prior art for the ‘263 patent that was in the stream of commerce…
It appears Lotus Notes 4.0 had this feature in January 1996:
http://books.google.com/books?id=zT4EAAAAMBAJ&pg=PA41&lpg=PA41&dq=url+recognition&source=bl&ots=igWdqkub6k&sig=n-4WdDFD3Gkz0cZO6hWyX9b-kSM&hl=en&ei=ZgwjTpf0B474sAOCyJA6&sa=X&oi=book_result&ct=result&resnum=9&ved=0CFcQ6AEwCDigAQ#v=onepage&q=url%20recognition&f=false
Hopefully, Apple applied for the patent sometime after January 22.
@Patrick, I’m a concerned citizen. I have no contacts within HTC to speak of.
Since they have no contact email for their legal department, I have sent a snail-mail letter to HTC’s headquarters advising them of the Lotus Notes prior art and pointing them at both the archived InfoWorld article and this comment thread.
@Patrick Maupin: Mostly just normals. A few programmers.
@Max E:
Not just text editors. GNOME Terminal and Konsole have had this feature going all the way back to the very earliest versions of each. Most of your IM clients — including AOL’s. Microsoft might have something to say, too, since Office has had it since maybe Office 95 or 97. Lots of prior art here, though I’m not sure how many of them actually pre-date the filing date, although all the examples I’ve given are contemporary.
Though one example I know for sure predates the filing date would at be at least one of the DOS modem communications packages. I’m not sure which of ones would skim phone numbers for you automatically, but I think at the very least Telix ca. 1989 and {COMMO} ca. 1989 would do it. Quite possibly Qmodem and ProComm+. And possibly FrontDoor’s integrated term. I can’t, unforunately, recall which programs had which features since during that time period I had tried and used them all.
At the very least it fails the obviousness test.
Text-to-Speech systems have to recognize telephone numbers, prices and addresses.
I think Festival could do it from the start, that would be half way the nineties. And they most certainly were not the first.
This paper by Richard Spraot from 1998 might contain relevant work and references ()sorry, on the road myself with very little time)
Emu: an e-mail preprocessor for text-to-speech
http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=738941
I would suggest that HTC gets in contact with Richard Spraot (he might be at OGI, Portland). He is a leading expert in the field. They might also want to have a talk wit Alan Black of CMU Pittsburg (one of the originators of Festival). He might have relevant information about early implementations.
Addendum to my last post: It wasn’t Telix or {COMMO} — I just downloaded and tried those in DOSEMU. Terminate has some nice functionality in this regard, though, but it doesn’t really automatically find phone numbers. Yet, I’m pretty sure some comm program was able to do that, ISTR it could even highlight them automatically in the scrollback buffer. I’m just not sure which one.
As I mentioned earlier, I may actually have coded some prior art for the ’263 patent (on real time signal processing). I found the email addresses of Mot’s lawyers and sent them some info. It will be interesting to see how that works.
As far as the other (‘647) patent, did anybody else read it? It has to be narrowed significantly on re-exam — claim 1 seems to claim the entire web, and it was filed in 1996.
@Mr. Maupin:
Sorry to get things so off course, but I was really more (initially) interested in Devlin’s comments about “Ordinary people who become proficient in everyday math in their daily lives to the point where their accuracy rate is around 98% when they do it mentally in a real setting, find that their performance level drops to around 37% when they try to do it symbolically. ”
Now, at first I thought this meant that there was a study somewhere that showed “most” people could do arithmetic to ~98% accuracy until you did something like put it algebraically, or used something like a sigma or the pie symbol. Then I re-read it and realized that he was only referring to people who do some sort of every day math (say a carpenter, or a bank teller) get confused when you describe the problem to them symbolically.
I think he’s probably correct about this, and since I bought his book, we’ll see what he has to say, if I can manage to read more than a couple pages at a sitting.
Oh, and don’t be too hard on Devlin, he’s a prof at Standford, and doesn’t appear to have started a multi-billion dollar company. The ignominy must be crushing.
@William O. B’Livion:
Actually, I thought the link you posted was very much on-point. Completely relevant to the issue of whether tau might be better than pi simply because it’s a more understandable way of thinking about and teaching the concepts.
I’m quite sure he’s correct about this issue. Symbols are crude avatars of concepts inside our minds. They help us compensate for our limited internal working memory. There is no question that many hard mathematical problems require more working memory than most people have, so in order to do what Mr. Devlin thinks of as serious math, (today) we have to become proficient at manipulating these symbolic avatars. He mentions something “better” that he will discuss more in an upcoming column — that might actually be worthwhile if true.
But just because the carpenter is not working with problems that require a large memory working set doesn’t mean that he isn’t doing “real” mathematics. In fact, I would wager that for small math word problems that involve real numerical multiplication and division, the carpenter will usually have the problem solved before the average mathematician has assigned variable names.
This is a corollary to my observation that it’s often the worst programmer who displays the highest proficiency with the mechanics of the debugging tools. It’s partly a chicken and egg feedback loop. Not thinking things through very well means they need a debugger to help, which teaches them they don’t need to think things through.
Heh. Maybe my ignominy is why I’m so hard on him…
You might enjoy “a mathematician’s lament”
http://www.maa.org/devlin/LockhartsLament.pdf
Funny, true, and instructive.
@Winter:
and the follow-up:
anti-matters.org/articles/93/public/93-86-1-PB.pdf
I don’t see the problem with treating multiplication as “repeated addition” and addition as “repeated counting”; that’s how I learned each of the operations.
And I don’t even see how it’s just a positive-integer thing.
Let’s consider
12 × x, concretized in terms of cooking for a large number of people a dish that includes eggs. If I tell someone I want to get12 × 2.5eggs, (once they understand what .5 means) they know I want to add two full cartons of eggs, plus another half carton, or a total of 30 eggs, to this big-ass mixing bowl here.HTML fail
> I don’t see the problem with treating multiplication as “repeated addition” and addition as “repeated counting”
And exponentiation as “repeated multiplication” (another no-no according to Mr. Devlin)
> And I don’t even see how it’s just a positive-integer thing.
Technically, it’s not. But every increment past positive integers is a teaching moment, or according to Mr. Devlin, an explanation that you lied before.
@The Monster:
Nobody says they want 12*2.5 eggs. Well, no one except a geek like you or me. They say they want “two and a half dozen eggs.” And no, this is not really a minor point.
@Morgan:
What if the dish called for 2.5 eggs/person, and it was being cooked for 12 people?
In any case, the problem, such as it is, with extending the counting paradigm to multiplying with fractions, is that the actual definition of teaching multiplication gets a bit loosey-goosey and circular. (Which, I’m sure, is how it goes in real life in any case.)
If somebody already knows the meaning of “half” well enough to give you half the eggs out of a carton, then they might already know how to multiply by that particular fraction. Multiplication is an organic concept that grows from a lot of different directions.
But if someone only knows “half” well enough to count twice on each egg for a total of 12 counts, then you can certainly easily extend from multiplying natural numbers to multiplying a natural number by a (simple) fraction. The next step — multiplying two fractions together — might be a bit trickier.
@Patrick Maupin: 0.5 * 0.25 is a a “quarter of a half” or, alternative “half of a quarter”. That’s an eighth or 0.125.
It’s all a matter of terminology.
@Morgan:
Oh, I agree, but you probably don’t want to start out there. Small steps. Multiply whole numbers, then a whole number by a fraction, etc…
There is nothing wrong with viewing natural number multiplication as repeated addition–every time we extend the natural numbers (to integers, fractions, reals, complex numbers) we need to change the view of multiplication, because the nature of multiplication needs to be extended to things it had never been extended to before.
I say this as a mathematician, and as a specialist in commutative algebra. Abstract Algebra is the art of creating new number systems from old ones. Thus, we get multiplication that is “repeated addition”, or “scaling”, or “rotating and scaling”, or “meaningless” Just what is multiplication supposed to mean in a polynomial ring, for example? Or Integers mod p? Or in matrices? For that matter, why are we limiting our view of multiplication as “scaling”? When we’re talking about integers and real numbers, multiplication can also be thought of as area–and this idea can provide just as much understanding, if not more, than multiplication as “scaling”.
When I first encountered the idea that multiplication as repeated addition was “wrong”, I sort-of accepted it–but the more I thought about it, the more I realized that there’s nothing wrong with the initial idea that multiplication is “repeated addition”; indeed, it provides us with opportunities to introduce the idea of “extending” a number system, to make the system more “complete”.
For the record, I’m a Tauist, and was even before the Tau Manifesto was written–I attended a presentation by Bob Palais himself years ago, before anyone recommended tau for the symbol. To me, the “Pi Manifesto” seemed to be a “but, but, but…pi is really right!” When I was working in complex analysis, equations were littered with “2*pi” factors.
1. “If a chicken and a half lays an egg and a half in a day and a half, how many eggs does one chicken lay in one day?” As SF writer Samuel R. Delany observes, an awful lot of people answer “one.”
2. ? vs. ?: too late. Lower-case-tau has done been claimed and it’s one of those symbols that’ll go home with any ol’ profession or science that’ll slip it a little formula. Little pi, on the other hand, is as loyal and pure as a Victorian heroine; there’s never a question what it represents. If symbols are tools, “?” is one of those hammer/pliers/screwdriver things with a bottle opener and pin punch on it.
3. Please Don’t Scare The Horses: as a radio-minded teen, the first time I stumbled over angular frequency represented as “?” instead of “2?f,” I did a double-take. It’s not merely size of the symbol set, it’s familiarity.
@Roberta X:
If I had a chicken that laid 2/3 of an egg every day, it would be chicken ‘n dumplings for supper…
@Roberta X:
BTW, while we’re on the subjects of puzzles and pedagogy, I think that if you’re presenting a puzzle like that, you should be prepared for some pushback because you will never find data in real life that got acquired as, or simplified to, 1.5 chickens, 1.5 days, 1.5 eggs. That’s just not gonna happen. Ignoring the stupidity of collecting 1.5 eggs from 1.5 chickens, 1.5 days doesn’t match any natural cycle, and it’s too short for any kind of meaningful average. So while it might be fun for some as a brain-teaser, it has to be presented and taught and dealt with in a very playful manner.
But, if you do decide to present a playful puzzle like this, be sure you get it right yourself. This starts with making sure the problem is not ambiguous.
One thing that really pissed me off about 15 years ago was “the smartest person in the world’s” treatment of a puzzle. A girl wrote to “Ask Marilyn” with a problem posed by her algebra teacher:
The girl who wrote to Marilyn was puzzled because she thought the probabilities were the same. Marilyn never explained that the reason she was puzzled was because the problem was ambiguous. Instead, Marilyn sided with the algebra teacher’s explanation, and when thousands of people wrote in to explain that was wrong, Marilyn never addressed the ambiguity. Instead, this stuck-up, know-it-all, had-to-be-right-at-all-costs major bitch created an unambiguous variant of the problem that she posed as a poll and used the result to “prove” the correct answer.
But that “scientific” explanation completely ignores some (maybe most) of the usual ways that you might find out “that at least one of the woman’s children is a boy.” E.g., if you meet a woman with a child in tow and get to talking to them, it might come up in casual conversation that they have one more child. The a priori probability that the other child is a boy is approximately 1/2, not approximately 1/3, regardless of the gender of the child you see standing in front of you with the woman.
So when the “smartest person in the world” teaches something that goes against the grain of common sense without explanation, and then creates a bogus poll to prove that she was right, all that does is teach clever people who were happily and correctly reasoning out things for themselves that you have to suspend reality in order to get math results that “experts” will consider to be correct.
@Alpheus:
> For that matter, why are we limiting our view of multiplication as “scaling”?
I dunno. That’s what the “expert” claimed it had to be, which is why I thought the car multiplication question was so useful.
The quality of math education in this country is quite disgusting. It’s two eggs every three days, of course. Patrick’s 2/3rds of an egg is, of course, kind of silly because there is no such thing.
> Patrick’s 2/3rds of an egg is, of course, kind of silly because there is no such thing.
But my whole point was that the entire puzzle was kind of silly because, unless you’re really stupid or deliberately trying to confuse, you don’t wind up with the data in that form.
I mean, c’mon. “If you have one and a half chickens, how many eggs should they lay in a day and a half?”
Well, for a start, I don’t think half a chicken will be laying any eggs, today, tomorrow, or ever…
The other two halves might stand a chance of some egg-laying, unless of course the owner started with halves he bought at the grocery store, and just glued two of them together.
@ Patrick, who wrote: “BTW, while we’re on the subjects of puzzles and pedagogy, I think that if you’re presenting a puzzle like that, you should be prepared for some pushback because you will never find data in real life that got acquired as, or simplified to, 1.5 chickens, 1.5 days, 1.5 eggs.”
Elsewhere, playfulness and ambiguity are mentioned. I was, however, quoting a famously dyslexic SF writer and it was not so much a story problem as an example of the manner in which presentation affects processing: handed three equal quantities in what look like a nice, neat form, the quick & lazy answer is wrong. The key to solving it is either immediate abstraction to numbers, which results in a hapless chicken trying to lay a fractional egg -or- Greywolfe’s method of working it in the real-world achievable terms of (integer) chickens laying (integer) eggs in (integer) days.
Where I am going with this is that I know what pi is; given some string, nails and yardstick, I can go experimentally derive it in the alley and if I am very lucky, I might even be as close as 22/7. But tau? Even the need for it isn’t especially obvious at the wood barn, ziggurat and taxing-by-enclosed-area level where most folks operate. Give the tax assessor tau and watch your taxes go up; or maybe down, if you can convince her that six-and-a-quarter is close enough.
(Daggone it, I most carefully pasted them-there Greeky letters last time I was here and today my browser claims it ain’t never heard of ’em. Diogenes, scooch over and spare a light, I’m lookin’ for an honester browser).
Agreed. As I mentioned in earlier posts, π can be useful for real-world scenarios involving diameters, which crop up in the alley of which you speak, and τ is useful for real-world scenarios involving radii, which crop up in fields like electrical engineering. And as far as your “please don’t scare the horses” argument, a lot of horses who have to do serious DSP work will welcome tau with open arms, even if it isn’t declared “fundamental.”
BTW:
> handed three equal quantities in what look like a nice, neat form, the quick & lazy answer is wrong.
See, that’s where my mileage varies. My initial reaction upon seeing that problem was “Why the f*** would anybody do that? They’re just f***in’ with me.” If you wanted to sneak up on me with that sort of problem, you’d have to catch me in a distracted mood, and with much more plausible-looking data — say, 2 chickens, 2 days, 2 eggs. I’m not saying I’d fall for it, but you might stand a chance. But in any case, half an egg a day per chicken is much more plausible — at least I can glue two halves together. Gluing two 2/3rds together will make for a really funny looking egg, probably hard to sell.
> (Daggone it, I most carefully pasted them-there Greeky letters last time I was here and today my browser claims it ain’t never heard of ‘em. Diogenes, scooch over and spare a light, I’m lookin’ for an honester browser).
THREAD WINNER!
Yours,
TOm
@Roberta X 9:23am.
To be fair to tau, pi isn’t as faithful as you think. In abstract algebra and topology, for example, pi is often used to represent a “projection map”; pi_0 (pi-sub-zero) is also the “fundamental group” of topology (and pi_i represents other such groups)–although, to be fair, the subscript in this case helps to differentiate the constant from the group.
I can’t find the example, but I remember an electrical engineering equation presented where pi meant the circle constant in one part of the equation, and was something else in the other part, but both symbols looked identical. Thus, tau isn’t the only constant that’s going to have symbol-collision issues. When you get deep enough in mathematics–or any math-heavy field of endeavor, I suppose–you get to the point where you need so many variables, you’ll just use whatever comes to mind. Mathematicians can run out of variables very quickly!
For what it’s worth, I remember using tau a lot in differential geometry, and I don’t remember the circle constant coming up in that field at all. It’s been several years since I took that class, though, or even read the textbook, so my memory on that may be a little off…
>For what it’s worth, I remember using tau a lot in differential geometry, and I don’t remember the circle constant coming up in that field at all.
My main association for τ is tau-time in relativity theory.
Many people considered (and probably stil consider) Godel to be a prime example
of a Platonist.(Just Google “Godel Platonist” for starters) Based on what I’ve read
about him, I don’t believe he would have construed the import of this theorems
anything like the way you do.
>I don’t believe [Gödel] would have construed the import of this theorems anything like the way you do.
You are quite right. He didn’t.
However, most others in the field did. Among metamathematicians it is considered a fine irony that Platonism was given its quietus by a Platonist.
Perhaps you can elucidate the common wisdom among metamathematicians that somehow
escaped poor Godel. I doubt anything purely logical would have escaped him though,
I suspect there is something else at work.
After more searching & reading I found various interesting papers online,
here’s a couple:
http://www.hss.cmu.edu/philosophy/techreports/106_Awodey.pdf
http://jvrosset.free.fr/Goedel-Proof-Truth.pdf
Best I can tell Godel Incompleteness cannot be unequivocally said to confirm
nor deny whatever view you start with regarding mathematical truth. For Godel’s
own typically honest assessment, try Googling “It is easy to allege very weighty
and striking arguments in favor of my views”.
I teach trigo and mensuration. Pi is indeed a pedagogical disaster. A thought process which should have been done in one step is unnecessarily burdened.
The factor one half is native to areas. Because it is native to the integration of linear functions which result into to quadratic function with factor of 1/2. This is further buttressed by the foundational role of triangles in measurements. A figure can only be measured if it can be sliced into solvable triangles, in some cases infinite slices. And the areas of triangles always have a factor of 1/2. So n-gons can be sliced into n triangles. And a circle can be sliced into infinite number of isosceles triangles with two congruent sides equal to its radius. So (n slices) (1/2 ) ((tau r)/n) r is the is the area of the circle as n approaches infinity.
The best approach in the adoption of tau is to request calculator manufacturers to put that constant in new calculators. I for one will not use pi in trigo and other subjects I teach if tau is in the calculator. Indeed pi is pedagogical disaster. Just imagine the routing process going through the minds of those young uns to visualize pi/3 radian of turn. I can’t even make the routing in 10 seconds myself.
More power to your advocacy. Count me in. Lemme have share that tau (the whole thing).