# The Simple Math of Everything

I am not a pro­fes­sional evolu­tion­ary biol­o­gist. I only know a few equa­tions, very sim­ple ones by com­par­i­son to what can be found in any text­book on evolu­tion­ary the­ory with math, and on one mem­o­rable oc­ca­sion I used one in­cor­rectly. For me to pub­lish an ar­ti­cle in a highly tech­ni­cal ev-bio jour­nal would be as im­pos­si­ble as cor­po­ra­tions evolv­ing. And yet when I’m deal­ing with al­most any­one who’s not a pro­fes­sional evolu­tion­ary biol­o­gist…

It seems to me that there’s a sub­stan­tial ad­van­tage in know­ing the drop-dead ba­sic fun­da­men­tal em­bar­rass­ingly sim­ple math­e­mat­ics in as many differ­ent sub­jects as you can man­age. Not, nec­es­sar­ily, the high-falutin’ com­pli­cated damn math that ap­pears in the lat­est jour­nal ar­ti­cles. Not un­less you plan to be­come a pro­fes­sional in the field. But for peo­ple who can read calcu­lus, and some­times just plain alge­bra, the drop-dead ba­sic math­e­mat­ics of a field may not take that long to learn. And it’s likely to change your out­look on life more than the math-free pop­u­lariza­tions or the highly tech­ni­cal math.

Not Ja­cobean ma­tri­ces for fre­quency-de­pen­dent gene se­lec­tion; just Hal­dane’s calcu­la­tion of time to fix­a­tion. Not quan­tum physics; just the wave equa­tion for sound in air. Not the max­i­mum en­tropy solu­tion us­ing La­grange mul­ti­pli­ers; just Bayes’s Rule.

The Sim­ple Math of Every­thing, writ­ten for peo­ple who are good at math, might not be all that weighty a vol­ume. How long does it take to ex­plain Bayes’s Rule to some­one who’s good at math? Damn would I like to buy that book and send it back in time to my 16-year-old self. But there’s no way I have time to write this book, so I’m toss­ing the idea out there.

Even in read­ing pop­u­lar works on sci­ence, there is yet power. You don’t want to end up like those poor souls in that re­cent in­ter­view (I couldn’t Google) where a well-known sci­en­tist in field XYZ thinks the uni­verse is 100 billion years old. But it seems to me that there’s sub­stan­tially more power in push­ing un­til you en­counter some ba­sic math. Not com­pli­cated math, just ba­sic math. F=ma is too sim­ple, though. You should take the high­est low-hang­ing fruit you can reach.

Yes, there are sci­ences whose soul is not in their math, yet which are nonethe­less in­cred­ibly im­por­tant and en­light­en­ing. Evolu­tion­ary psy­chol­ogy, for ex­am­ple. But even there, if you kept push­ing un­til you en­coun­tered equa­tions, you would be well-served by that heuris­tic, even if the equa­tions didn’t seem all that en­light­en­ing com­pared to the ba­sic re­sults.

I re­mem­ber when I fi­nally picked up and started read­ing through my copy of the Feyn­man Lec­tures on Physics, even though I couldn’t think of any re­al­is­tic ex­cuse for how this was go­ing to help my AI work, be­cause I just got fed up with not know­ing physics. And—you can guess how this story ends—it gave me a new way of look­ing at the world, which all my ear­lier read­ing in pop­u­lar physics (in­clud­ing Feyn­man’s QED) hadn’t done. Did that help in­spire my AI re­search? Hell yes. (Though it’s a good thing I stud­ied neu­ro­science, evolu­tion­ary psy­chol­ogy, evolu­tion­ary biol­ogy, Bayes, and physics in that or­der—physics alone would have been ter­rible in­spira­tion for AI re­search.)

In academia (or so I am given to un­der­stand) there’s a huge pres­sure to spe­cial­ize, to push your un­der­stand­ing of one sub­ject all the way out to the fron­tier of the lat­est jour­nal ar­ti­cles, so that you can write your own jour­nal ar­ti­cles and get tenure. Well, one may cer­tainly have to learn the far math of one field, but why avoid the sim­ple math of oth­ers? Is it too em­bar­rass­ing to learn just a lit­tle math, and then stop? Is there an un­writ­ten rule which says that once you start learn­ing any math, you are obli­gated to finish it all? Could that be why the prac­tice isn’t more com­mon?

I know that I’m much more em­bar­rassed to know a few sim­ple equa­tions of physics, than I was to know only pop­u­lar physics. It feels wronger to know a few sim­ple equa­tions of evolu­tion­ary biol­ogy than to know only qual­i­ta­tive evolu­tion­ary biol­ogy. Even men­tion­ing how use­ful it’s been seems wrong, as if I’m boast­ing about some­thing that no one should boast about. It feels like I’m a dilet­tante—but how would I be dilet­ting less if I hadn’t stud­ied even the sim­ple math?

• But there’s no way I have time to write this book, so I’m toss­ing the idea out there.

Would you have time to start a wiki whose pur­pose was to be ed­ited into a book, coau­thored by dozens of con­trib­u­tors, who can ex­plain the ba­sic sim­ple math of their field to non-math-pho­bic laypeo­ple? (This is differ­ent from just scrap­ing Wikipe­dia; these would be tar­geted ar­ti­cles, per­haps some in­vited ones...) Of course that could end up tak­ing more time due to the in­fa­mous herd­ing cats prob­lem. But I’d love to have that book to read on the BART train.

• I don’t think most peo­ple feel more ashamed of know­ing a lit­tle than know­ing noth­ing; they just don’t try. But, Eliezer’s shame re­minds me of the story where Feyn­man is hav­ing trou­ble learn­ing some­thing, and his wife tells him to read like a be­gin­ner again. I be­lieve it is a com­mon spec­u­la­tion that peo­ple avoid learn­ing new things to avoid feel­ing like a be­gin­ner.

• I re­mem­ber reach­ing ex­actly this point and mak­ing ex­actly this wish many years ago. I tried to learn as many fields as I could by read­ing in­tro­duc­tory text­books, and most of those texts avoid any math. I thought that a text that was will­ing to use sim­ple math could teach me a lot more a lot faster. My the­ory was that there were too few peo­ple who could han­dle sim­ple math and would want to learn many fields to sup­port the book. But I’d love to be shown wrong.

• A bet­ter way of look­ing at it may be as Math­e­mat­ics for Un­der­stand­ing, as op­posed to maths for re­search, in­stead of Sim­ple Math.

• What I find em­bar­rass­ing about know­ing just a lit­tle bit about a sub­ject is that out­side of a for­mal class, there are few places to talk about it; par­tic­u­larly, few places to talk about it with peo­ple who will bring your fur­ther to­ward un­der­stand­ing what you’ve learned. If you learn a lit­tle bit of the math­e­mat­ics of a sub­ject, you’re not in­ter­est­ing to the spe­cial­ists, and most oth­ers won’t be in­ter­ested in the sub­ject at all.

It seems eas­ier to find a com­mu­nity around learn­ing things that are less aca­demic sub­jects, where you’ll gen­er­ally learn them in an in­for­mal struc­ture any­how—cook­ing, crafts, for­eign lan­guages.

(I do like the idea of The Sim­ple Math of Every­thing...)

• Hi, I’m a lurker on this site. I think this is a brilli­ant idea. I’ve just set up a wiki at http://​​scratch­pad.wikia.com/​​wiki/​​The_Sim­ple_Math_of_Everything

• This page has been deleted. The dele­tion and move log for the page are pro­vided be­low for refer­ence.

• I agree about the use­ful­ness of a ba­sic tech­ni­cal un­der­stand­ing of as many fields as pos­si­ble. As for the push to spe­cial­ize in academia- well, it’s com­pli­cated. I’m not a pro­fes­sor, I’m a grad stu­dent, but here’s my ex­pe­rience. If you’re in one of the rel­a­tively “pure” dis­ci­pline- physics, com­puter sci­ence, and so on- the push to spe­cial­ize is very real, as is the push to fo­cus on what ev­ery­one else (in­clud­ing grant­ing agen­cies) thinks is “hot.” But there is a lot of multi-dis­ci­plinary work go­ing on, an in­creas­ing amount re­ally. Trou­ble is, that quickly be­comes a new dis­ci­pline in its own right. My alma mater now has 5 differ­ent biol­ogy ma­jors, each of them in­ter­dis­ci­plinary in in­ter­est­ing ways. My own field- ma­te­ri­als sci­ence- en­com­passes the study of solids and liquids. Me­tals, al­loys, ce­ram­ics, ox­ides, semi­con­duc­tors, polymers, and even biolog­i­cal ma­te­ri­als. It can’t be done un­less you un­der­stand or­ganic and in­or­ganic chem­istry, crys­tal­log­ra­phy (ap­plied group the­ory, re­ally), physics (clas­si­cal- strain fields, shear­ing forces; and quan­tum- bloch waves, elec­tronic band struc­ture), and enough com­puter sci­ence to right some ba­sic simu­la­tions. You end up with pro­fes­sors work­ing in fields that didn’t ex­ist when they started out. So they keep tak­ing classes and read­ing each other’s books.

• A lit­tle knowl­edge can be more dan­ger­ous—and em­bar­rass­ing—than com­plete ig­no­rance.

Yes. As a math pro­fes­sor, I sort of agree and sort of dis­agree with this post. On the one hand, peo­ple have lots of mi­s­un­der­stand­ings about math, as peo­ple like John Allen Paulos have writ­ten. But on the other hand, it’s NOT true that ev­ery­thing has a sim­ple math­e­mat­i­cal model. Often math­e­mat­i­cal mod­els that might be use­ful in physics are not es­pe­cially use­ful el­se­where, and even more of­ten the most im­por­tant thing is not the model’s pre­dic­tions, but the er­rors.

Look at the So­cial Se­cu­rity model, for ex­am­ple. It’s in­cred­ibly un­re­li­able, be­cause it makes long-time pre­dic­tions based on a sin­gle pa­ram­e­ter (av­er­age growth of GNP) which is as­sumed to be con­stant over 40 years. And the differ­ence in pre­dic­tions by chang­ing this widely vary­ing num­ber is on the or­der of 10-20 years.

But the prob­lem is that a few peo­ple think they know the math here and think they un­der­stand the situ­a­tion com­pletely be­cause of it. In fact they know a tiny bit of math (or trust that other peo­ple know the math), and end up do­ing in­cred­ibly stupid things be­cause of it. If they ac­tu­ally knew more, they would be a lot more care­ful with things like per­sonal ac­counts and such. In­stead we trust a few poli­ti­cal ap­poin­tees, pro­cess a cou­ple of the num­bers in­volved, and base ev­ery­thing on that.

And if you dis­agree with me about per­sonal ac­counts on So­cial Se­cu­rity or some­thing, and just think I’m a liberal who shouldn’t be taken se­ri­ously, com­pare the Dooms­day ar­gu­ment http://​​en.wikipe­dia.org/​​wiki/​​Dooms­day_ar­gu­ment. It uses statis­tics (which most peo­ple don’t un­der­stand) to make a triv­ial pre­dic­tion with ab­surd con­se­quences that gets taken se­ri­ously. Peo­ple with a lit­tle un­der­stand­ing of statis­tics will take it se­ri­ously, but peo­ple who ac­tu­ally un­der­stand the limi­ta­tions of statis­tics will re­al­ize it’s ridicu­lous.

• But the prob­lem is that a few peo­ple think they know the math here and think they un­der­stand the situ­a­tion com­pletely be­cause of it. In fact they know a tiny bit of math (or trust that other peo­ple know the math), and end up do­ing in­cred­ibly stupid things be­cause of it.

Agreed, but peo­ple with enough ex­pe­rience of the limits of sim­ple math­e­mat­i­cal mod­els in one field are less likely to make that mis­take in other fields.

A hy­po­thet­i­cal “The Sim­ple Maths of Every­thing” text­book should in­clude warn­ings about the limits of the mod­els, and a few mem­o­rable ex­am­ples of how those mod­els go wrong.

• If you ever get as se­ri­ously cu­ri­ous about elec­tron­ics as you were about physics, look at Horow­itz and Hill, The Art of Elec­tron­ics. Very very use­ful for some­one who already knows the math and wants to un­der­stand elec­tron­ics prin­ci­ples and the prac­ti­cal­ities of one-off dis­crete cir­cuit de­sign.

• While sim­ple “me too”ing is gen­er­ally bad ne­ti­quette, I have to say that The Sim­ple Math of Every­thing is a just plain fan­tas­tic idea.

• My guess is that most peo­ple sim­ply don’t know that know­ing the math is im­por­tant to un­der­stand­ing a sub­ject. Un­til you have some tech­ni­cal un­der­stand­ing of a sub­ject it may seem that a non-tech­ni­cal un­der­stand­ing is all there is.

• A lit­tle knowl­edge can be more dan­ger­ous—and em­barass­ing—than com­plete ig­no­rance.

• Has there been any progress to­wards this idea? I as well think it would be a fan­tas­tic book and would love to read it

edit: I see there’s a wiki page re­gard­ing this idea, with some links

• The dan­gers of a “lit­tle learn­ing” are eas­ily offset by point­ing out the ways the rele­vant “sim­ple math” fails in a given case. Cf. Feyn­man’s (for ex­am­ple) use of analo­gies. He’d state the anal­ogy, then point out the ways in which the anal­ogy is wrong or mis­lead­ing, the spe­cific fea­tures that fail to map, etc. This strat­egy gets you the ped­a­gog­i­cal benefits of struc­ture map­ping while min­i­miz­ing the risk (that Bill Swift warns against, supra) that a lit­tle learn­ing will be mis­taken for a great deal.

• There are some laud­able at­tempts for such a book by a few peo­ple, the first one com­ing to mind is “the com­pu­ta­tional beauty of na­ture”. Although it con­tains only a few fields, it’s still a great book for the “not-afraid-of-a-few-ba­sic-equa­tions” crowd. Wish there were more books like that.

• Add the in­for­ma­tion to Con­nex­ions. (http://​​cnx.org/​​) It seems built for just such a pur­pose and was high­lighted in one of the TED talks a year or so ago if any­one wants to go watch a video overview.

• Beau­tiful idea!

Is a Wiki sep­a­rate from Wikipe­dia needed?

Similar prob­lem: One thing I run in to of­ten on Wikipe­dia is en­tries that use the field’s par­tic­u­lar math­e­mat­i­cal no­ta­tion for no rea­son other than par­tic­u­lar sym­bols and ex­pres­sions are the jar­gon of the field. They get in the way of un­der­stand­ing what the en­try is say­ing, though.

Similar prob­lem is there seem to be aca­demic pa­pers that have prac­ti­cal ap­pli­ca­tions and yet the pa­pers are writ­ten to be as un­clear as pos­si­ble—per­haps to take on that “im­por­tant” sheen, per­haps sim­ply be­cause the au­thors are deep in their own jar­gon and as­sume all read­ers know ev­ery­thing they know. Con­sider pa­pers in the AI field. :)

• Pete: I was just think­ing the same thing, that we ought to start a wiki to do this pro­ject. Ques­tions do come up though like “where ought one draw the line be­tween the sim­ple and non­sim­ple”? This ques­tion re­lates even ti billswift’s com­ment about the name.

For in­stance, in physics, ought we in­clude Hamil­ton’s equa­tions/​the hamil­to­nian? There’s cer­tainly un­der­stand­ing to be found by con­sid­er­ing a sys­tem in those terms. But de­riv­ing those and so on prob­a­bly is a bit deeper than what one might want to con­sider “easy math”… or maybe not. Those are in some ways the start­ing point that leads to the deep stuff.

There’s prob­a­bly analo­gous ques­tions in other fields. So we have to de­cide what we’re go­ing to con­sider the “easy” math.

• “where ought one draw the line be­tween the sim­ple and non­sim­ple”?

My sug­ges­tion would be not to draw the line… but to grade things on how hard they are (fun­da­men­tal, ba­sic, in­ter­me­di­ate...).

That way, any­body can start, and can stop at any time they want to...

• I’ve been read­ing a book similar to what you have in mind I think. It’s “Math­e­mat­ics: From the birth of num­bers” (http://​​www.ama­zon.com/​​Math­e­mat­ics-Birth-Num­bers-Jan-Gul­lberg/​​dp/​​039304002X). It starts very ba­sic but cov­ers all sorts of ad­vanced top­ics. It’s de­signed for some­one with no higher math learn­ing. I’m about 1/​​4 of the way through it and so far very im­pressed.

• First off, that book looks won­der­ful. It looks, just from the de­scrip­tion, like it goes deeper into Math, rather than cov­er­ing the math of other fields. As delight­ful as Math can be, I’d be much more in­ter­ested in hav­ing a primer on the math of all sorts of other things.

• Dou­glas, if all you say is “some cats have more ba­bies than other cats” then you have missed out the key el­e­ment of her­i­ta­ble vari­a­tion and there­fore haven’t said any­thing about evolu­tion by nat­u­ral se­lec­tion.

• Here’s Han­son’s take on the Dooms­day ar­gu­ment:

http://​​han­son.gmu.edu/​​nodoom.html

• Link is 404. I can’t find Han­son’s ar­ti­cle el­se­where. I so hate link rot.

• Steve, would you care to elu­ci­date what’s ridicu­lous about the Dooms­day ar­gu­ment? I’d be es­pe­cially in­ter­ested in an ex­pla­na­tion based on the “limi­ta­tions of statis­tics” as op­posed to a hand-wav­ing ar­gu­ment. The Dooms­day ar­gu­ment strikes most peo­ple as ab­surd on its face, and yet it’s sur­pris­ingly re­sis­tant to re­fu­ta­tion. My own opinion is that it’s not ab­surd at all, and is among the ideas that re­veal a deep truth about re­al­ity.

• Didn’t Steven Hawk­ing say that his pub­lisher told him that ev­ery equa­tion he put in his book would halve the sales? So that’s why real math doesn’t make it into most pop­u­lar sci­ence books, one of the rea­sons there’s a band-gap be­tween nar­ra­tive sci­ence and pro­fes­sional texts. Would be nice to have this filled, I agree.

• A lit­tle learn­ing is not a dan­ger­ous thing to one who does not mis­take it for a great deal. William A White

Quoted in Ron­ald Gross’s In­de­pen­dent Scholar’s Hand­book. Which, un­for­tu­nately, is not par­tic­u­larly use­ful for tech­ni­cal fields.

• That’s a GREAT idea. I’ve been try­ing to do the same as Robin, but the availa­bil­ity of good text­books is some­what limited where I live (and they’re quite ex­pen­sive to im­port). A vol­ume con­tain­ing the in­tro­duc­tory math for many fields would make things much eas­ier, and I’d cer­tainly be buy­ing it.

• Well an­swered!

• Dou­glas, I’m not say­ing that there are cats that don’t have her­i­ta­ble vari­a­tion, any more than you’re say­ing that there are cats that don’t have vary­ing num­bers of offspring. I’m say­ing that the fact that cats have her­i­ta­ble vari­a­tion is just as rele­vant to evolu­tion as the fact that their num­ber of offspring varies.

• g—cats with­out her­i­ta­ble vari­a­tion? Where you get some of them?

• If what you’re propos­ing is like a “Ad­vanced Math­e­mat­i­cal Prin­ci­pals for Dum­mies”, I think you have a great idea.

You say you don’t have the time, but you could prob­a­bly put to­gether a few peo­ple to put some­thing to­gether. 4-5 peo­ple writ­ing two chap­ters. The “Dum­mies” folks would prob­a­bly pub­lish some­thing like that. I’d con­sider buy­ing it.

• The math of a sub­ject is only valuable when one un­der­stands the ba­sic ter­minol­ogy of the sub­ject. As Chris points out, know­ing when to use statis­tics (the ba­sic as­sump­tions and what the word ap­plies to) makes some­thing like the Dooms­day Ar­gue­ment good for a laugh. It is ridicu­lous. On evolu­tion­ary biol­ogy-- Evolu­tion is defined as ” any change in the fre­quency of alle­les within a gene pool from one gen­er­a­tion to the next.” This fre­quency changes with each birth. So to make the defi­ni­tion into reg­u­lar English we could say Evolu­tion is defined as “liv­ing things re­pro­duce” (the fact of evolu­tion). In mo­dem evolu­tion­ary ge­net­ics, nat­u­ral se­lec­tion is defined as “the differ­en­tial re­pro­duc­tion of geno­types (in­di­vi­d­u­als of some geno­types have more offspring than those of oth­ers)”. In English- some cats have more ba­bies than other cats. So the state­ment “It is a fact that some cats have more ba­bies than other cats,” would be the proof of evolu­tion by nat­u­ral se­lec­tion as the terms are cur­rently defined. Doesn’t that help more than a math­e­mat­i­cal equa­tion?

• Evolu­tion is defined as ” any change in the fre­quency of alle­les within a gene pool from one gen­er­a­tion to the next.” This fre­quency changes with each birth. So to make the defi­ni­tion into reg­u­lar English we could say Evolu­tion is defined as “liv­ing things re­pro­duce” (the fact of evolu­tion).

This doesn’t fol­low.

• Well, the ob­vi­ous point is that the Coper­ni­can Prin­ci­ple is fre­quently wrong. The An­thropic Prin­ci­ple does a fairly good job at point­ing out the weak­nesses of the CP, to start with, and re­mem­ber­ing that all else is rarely equal takes care of most of the rest.

• Now will some­one set up a fu­tures mar­ket tied to the pub­li­ca­tion of a book with that ti­tle by a non-van­ity press within the next 18 months?

• I’d sure as hell buy it (well given it was not pub­lished by Springer and priced ac­cord­ingly :P)!

• Hi Guys,

I am also a lurker/​ad­mirer of this site and I would love to have such a book! I will be watch­ing this topic and the wikipe­dia linked to, hop­ing some­thing comes of it. Even­tu­ally I will put up sim­ple neu­ro­science equa­tions.

• GRAET JOB BRO

• hi and u suck

• hi and u suck