Does Math Objectively Exist, or Is It a Human Creation? A New PBS Video Explores a Timeless Question

In a famous scene from Boswell’s Life of Samuel John­son, the biog­ra­ph­er and his sub­ject come to dis­cuss the bizarre the­o­ries of Bish­op Berke­ley, who posit­ed that every­thing is immaterial—nothing has any real exis­tence; it’s all just ide­al con­cepts held togeth­er by the mind of God. If God should lose his mind or fall asleep or die, every­thing would fall to pieces or cease to exist. Boswell insists there’s no way to refute the idea. John­son, kick­ing a large stone with such force that his foot rebounds, cries, “I refute it thus.”

Johnson’s lit­tle demon­stra­tion doesn’t actu­al­ly refute Berkeley’s rad­i­cal ide­al­ism. It’s a conun­drum still with us, like Plato’s Euthy­phro stumper, which asks whether the rules gov­ern­ing human behav­ior exist inde­pen­dent­ly of the gods, who sim­ply enforce them, or whether the gods make the rules accord­ing to their whims. In oth­er words, is moral­i­ty objec­tive or sub­jec­tive?

A sim­i­lar prob­lem occurs when we con­sid­er the exis­tence of the rules that gov­ern phys­i­cal laws—the rules of math­e­mat­ics. Where does math come from? Does it exist inde­pen­dent­ly of human (or oth­er) minds, or is it a human cre­ation? Do we dis­cov­er math­e­mat­i­cal prob­lems or do we invent them?

The ques­tion has engen­dered two posi­tions: math­e­mat­i­cal real­ism, which states that math exists whether we do or not, and that there is math out there we don’t know yet, and maybe nev­er can. This posi­tion may require a degree of faith, since, “unlike all of the oth­er sci­ences, math lacks an empir­i­cal com­po­nent.” You can’t phys­i­cal­ly observe it hap­pen­ing. Anti-real­ists, on the oth­er hand, argue that math is a lan­guage, a fic­tion, a “rig­or­ous aes­thet­ic” that allows us to mod­el reg­u­lar­i­ties in the uni­verse that don’t objec­tive­ly exist. This seems like the kind of rel­a­tivism that tends to piss off sci­en­tists. But no one can refute either idea… yet. The video above, from PBS’s Idea Chan­nel, asks us to con­sid­er the var­i­ous dimen­sions of this fas­ci­nat­ing and irre­solv­able ques­tion.

Relat­ed Con­tent:

The Math in Good Will Hunt­ing is Easy: How Do You Like Them Apples?

Incred­i­ble Men­tal Math Gym­nas­tics on “Count­down”

Math Doo­dling

Josh Jones is a writer and musi­cian based in Wash­ing­ton, DC. Fol­low him at @jdmagness

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Comments (14)
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  • Laroquod says:

    Is there any way you could STOP repub­lish­ing every recent arti­cle, every day? Every day I go to check my RSS, and every sin­gle Open Cul­ture arti­cle has been repub­lished into the feed, even the ones I’ve read before. So even if you only pub­lished one arti­cle today, I have to flip through 20 Open Cul­ture entries today in order to find that sin­gle new arti­cle. The oth­er 19 are all reposts (updates? refresh­es? it mat­ters not) of all of the pre­vi­ous posts I have already read this week. Some of your arti­cles crop up as ‘fresh’ in your RSS feeds lit­er­al­ly a dozen times.

    This is a great site and all, but seri­ous­ly it’s not THAT great that I should wade through 95% reposts every sin­gle day. It all adds up to a giant waste of time. RSS is dif­fi­cult enough to man­age with­out you repost­ing *every­thing* *every* *day*.

    So, to sum up: RSS, you are doing it wrong. You are doing it very, very wrong. Thank you for your atten­tion. I give it about a week or two before my patience final­ly gives with your RSS feed. Maybe I will fol­low your Twit­ter account instead. (Of course I hard­ly ever read Twit­ter so you should not take this as a good alter­na­tive.)

  • Sha says:

    In ref­er­ence to RSS: I don’t have this prob­lem with my RSS read­er. Some­thing else may be going on?

  • JCA says:

    My RSS feed always dis­plays 2 of the same arti­cle. Which means the list is twice is long for the same thing. I agree with Laro­quod — some­thing is the mat­ter.

  • Rebecca W says:

    What rss read­er do you use? I recent­ly switched to Feed­ly and have had good luck. I also like the option­al mag­a­zine for­mat.

    PS: To the web­mas­ter: Could you enable the options on com­ments for “noti­fy me of new com­ments on this post” AND “noti­fy me of new posts by email”. Thanks!

  • Brannon says:

    This reminds me of the argu­ment that Stephen Hawk­ing makes in his “The Ele­gant Design” book, which, IMHO as a for­mer sci­en­tist and a cur­rent sci­ence teacher, is a full load of crock…

  • Andrew Suttar says:

    Maths is a lan­guage. It exists in our minds. It is used to artic­u­late rela­tion­ships. The rela­tion­ships may well be onto­log­i­cal though.

    And yet this blows my mind:

    Make a mobius strip. It now has 1 sur­face. Draw a sin­gle line down the mid­dle until it becomes a con­tin­u­ous loop. Cut along this line. The mobius now has 2 sur­faces.

    Make anoth­er mobius. Draw 2 par­al­lel con­tin­u­ous loops now. Cut down these loops in par­al­lel. It now makes a mobius looped through a dou­ble mobius, (i.e 3 sur­faces)

    Try it with 3, 4 and 5 par­al­lel lines and we have cre­at­ing the num­ber series.

    What is amaz­ing is that they are always looped through each oth­er and can be reassem­bled back into 1 mobius with only one sur­face.

    I won­der what Vihart has to say about all this .o?

  • Theis Peter says:

    All of human knowl­edge is approx­i­ma­tion — our attempt to create/impose a grid on some­thing wig­gly, organ­ic. Though we like to feel and think oth­er­wise, we most­ly fum­ble through the dark. Objec­tiv­i­ty seems to be the main issue — does it even exist? We humans attempt to define objec­tiv­i­ty, but we are attempt­ing to do so with our very lim­it­ed capac­i­ties. As Deleuze/Guattarri wrote, it´s like try­ing to “seize a blade of grass and hold fast to it when it begins to grow only from the mid­dle.” It´s not “you can´t step in the same riv­er twice,” instead it is prob­a­bly more like “You can´t step in the same riv­er once.”

  • lee says:

    This is a foist­ed dialec­tic. The dual­ism aris­es in fail­ing to see that every­thing is a process, includ­ing real­i­ty itself. Our con­scious expe­ri­ence exists only as a process, but as a prop­er­ty aris­ing out of its sub­strate.

    Maths are the same thing, only dif­fer­ent.

  • We needed says:

    Math exist . In the same way that psy­chol­o­gy exists.nHumans cor­re­lat­ed lan­guage to describe the inter­ac­tions of the uni­verse. for­mu­las and con­cepts are cre­at­ed by humans to com­mu­ni­cate abstract ideas of thing that actu­al­ly exist . nnUn­til Freud’s the­o­ries of the psych, we still would not ful­ly under­stand the brain’s full capac­i­ty. A biol­o­gist can tan­gi­bly note the aspects of the brain, but the mea­sur­ers of the brain’s full func­tion­ing capac­i­ty is paired with “meta-cog­ni­tion” or think­ing. The con­cept of a human’s brain capac­i­ty and how it impacts on their over all men­tal and phys­i­cal well-being is abstract (intan­gi­ble ) but real. nnMath­e­mati­cians are dis­cov­er­ing the inter-relat­ed­ness of things in the uni­verse, and devis­ing a for­mu­la to com­mu­ni­cate that uni­ver­sal dis­cov­er to all. n

  • asdf says:

    But… math does have an empir­i­cal com­po­nent. It’s called proofs. The results of a com­pu­ta­tion, which are iso­mor­phic to proofs, are unknown until after the com­pu­ta­tion has com­plet­ed.

  • Sam says:

    I’m a lin­guist writ­ing a paper on the the­o­ry of maths for my research at an Insti­tute of Neu­ro­science.

    Maths is a human lan­guage. Yes, maths with an ‘s’, I’m British.

    Human lan­guage is used to explain meta­physics.

    Meta­physics exists even with­out being per­ceived.

    There­fore math­e­mat­i­cal lan­guage is cre­at­ed (not real) but math­e­mat­i­cal mean­ing is dis­cov­ered (real).

    Ani­mals don’t under­stand the­o­ret­i­cal maths but many under­stand prac­ti­cal maths (like geo­met­rics and basic physics). Prey see­ing a preda­tor and run­ning fast in the oppo­site direc­tion? They’re using maths!

    The oppo­site of empiri­cism is ontol­ogy and the oppo­site of sci­ence isn’t faith, it’s log­ic.

    If a tree falls down and no human is there to hear it, does it make a sound? Yes! Because sound comes from sound waves which are elec­tro­mag­net­ic. Elec­tro­mag­net­ic activ­i­ty hap­pens even in the fur­thest reach­es of our Uni­verse. It’s a ques­tion of log­ic, not faith.

    Fic­tion is a form of meta­physics but it’s dif­fer­ent. You have to remem­ber how James Bond likes his drink, as opposed to under­stand it. It needs con­text. Pythago­ras’s the­o­rem can be under­stood, even out of con­text. That’s why it’s true.

    But it’s pos­si­ble that I’ve just lost my mind.

  • Mike Deutsch says:

    Sound is cre­at­ed by vibra­tions through a medi­um (i.e. Air, water etc.) that affect a hear­ing appa­ra­tus, as an ear. Elec­tro­mag­net­ic radi­a­tion (ocil­lat­ing elec­tric and mag­net­ic fields) will prop­a­gate thru a total vac­u­um (space) @ the speed of light (vs sound thru air @ Mach 1) solids (x‑ray, *lim­it­ed solids* & gam­ma *will pass thru your body* & oth­er solids), atmos­phere
    (vis­i­ble light, ultra­vi­o­let, radio waves) etc.

    Back to the ques­tion of a tree falling, if there was absolute­ly no enti­ty capa­ble of hear­ing or feel­ing the vibra­tions (no observ­er) includ­ing mea­sur­ing devices then how would you prove it makes a sound ? You might infer a sound is made with or with­out the observ­er
    based upon sci­en­tif­ic laws. The tree hits the ground pro­duc­ing vibra­tions in the earth, air or near­by water pro­duc­ing sound waves regard­less of an observ­er.
    How would you prove it in the absolute
    with­out a mea­sure­ment or more exact­ly an observ­er ? More to the point sound is only mean­ing­ful to those with the abil­i­ty to detect it. The sound of a falling tree is mean­ing­less when no thing hears it.

  • carolyn hoover says:

    The world was cre­at­ed in 6 days on the sev­enth God rest. So would­n’t you say math was here before sci­ence cause it points to the very first wk.the world was cre­at­ed.

  • nobody says:

    mobius strip has two sides and a tran­si­tion sec­tion wich per­mits the change from the two. It is a com­pos­ite fig­ure. If you take the two com­po­nent bands apart, you get two per­fect cla­si­cal 2 sided strips one of them with a twist in it. If you fol­low with a pen the sur­face of a mobius tape you end one side at the very moment you encounter the begin­ing of the trace in the oth­er side.From that moment you must trace with anoth­er col­or until you get to the begin­ing of the first trace to be fair with the log­ic of things. If you pass that point with­out mark­ing it and pre­tend­ing that you are in the same side, you are not fair. Take for exam­ple a paper band in wich every side is of one col­or and make with this band a mobius band. Has the col­or of the band changed in a way by the mere act of twist­ing and tap­ing the both sides. No. you get a point, (the union point) where the col­or of the band switchs. Here you have the proof that the moe­bius band is a fake con­cept, in front of you.

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