Universals and the Game of Life

The Problem of Universals

In this world we are sur­rounded by par­tic­u­lar things. A table, a chair, a house, a per­son… these are all indi­vid­ual par­tic­u­lar things. But, many of the par­tic­u­lar things around us have some­thing in com­mon. An apple is red and so might be a fire hydrant. A horse is fast and so is a race car. Napoleon was a per­son and so was his wife. Philoso­phers often use the term universals to describe these things that dif­fer­ent par­tic­u­lar objects have in com­mon, or in other words, any word that can be pred­i­cat­ed, or said, of mul­ti­ple things is a universal. That means just about any­thing, “Horse” in a uni­ver­sal, because it refers to a cat­e­gory of things called “hors­es”. “Hu­man” is a uni­ver­sal because it refers to a cat­e­gory of things called “hu­man­s”. “Red­ness” is a uni­ver­sal because it’s pos­si­ble for more than one thing to be the color red. “Large”, “short”, “white”, “table”, etc… just about any adjec­tive or noun that isn’t a proper noun is arguably a universal.

So, why do we care? Why invent a term like uni­ver­sal when we already have words like “noun” and “ad­jec­tive”? Well, the answer is because there is an old and long-­s­tand­ing debate in phi­los­o­phy as to the sta­tus of uni­ver­sals. Are uni­ver­sals “re­al”? That is, do things like, “or­ange­ness” or “hu­man­ness” inde­pen­dent of par­tic­u­lar orange things, or par­tic­u­lar humans? Some ancient philoso­phers such as hugely influ­en­tial Plato, would have emphat­i­cally argued “yes”.

Plato famously pos­tu­lated the World of Forms, a sep­a­rate world of pure forms upon which every­thing in our real, sen­si­ble, world is based. Forms in Pla­to’s jar­gon, is just another word for what we are here call­ing universals. At first blush, this idea seems a bit odd: why would some­one feel the need to believe that every­thing in the world is based on things in this sep­a­rate world of pure forms? The idea seems a bit unnec­es­sary. How­ev­er, there is a logic to the idea. Imag­ine, for the sake of argu­ment, a geo­met­ri­cal fig­ure, say a tri­an­gle. There are many things in the world that we can say are roughly triangular: pyra­mids, cer­tain road signs, Pic­cadilly Cir­cus, etc. Almost noth­ing in the world, how­ev­er, can be said to be a per­fect tri­an­gle. Rather, things in the real world some­times approx­i­mate a tri­an­gu­lar shape, so tri­an­gles as such don’t nec­es­sar­ily exist in the indi­vid­ual tri­an­gle shaped things in the world. Yet, we can still rea­son about per­fect tri­an­gles and know, for exam­ple, that all (Euclid­ean) tri­an­gles have three cor­ners whose angles sum up to 180°. If we can rea­son about tri­an­gles sep­a­rately from indi­vid­ual tri­an­gle shaped things, it stands to rea­son, so Plato sur­mised, that “tri­an­gle­ness” is some­thing sep­a­rate from those indi­vid­ual tri­an­gle shaped things, and must have its own sep­a­rate exis­tence. Pla­to’s approach to uni­ver­sals is called metaphysical realism; he believed that uni­ver­sals are real in the sense that they have their own sep­a­rate exis­tence from the par­tic­u­lar objects things that we expe­ri­ence in the real world.

Other philoso­phers, such as Aristotle, take a bit more mod­er­ate approach to meta­phys­i­cal real­ism. For Aris­totle, there was no world of forms, rather, for him forms existed in the par­tic­u­lar objects which were instances of dif­fer­ent forms. So, for exam­ple, a horse was said to be an instance of the form of hors­e­ness, and the form of hors­e­ness was present in each indi­vid­ual horse. For Aris­totle, what made a horse, was that it instantiated the form of hors­e­ness. For Aris­totle, forms could also exist in the minds of peo­ple who com­pre­hended those objects. So, if you saw a horse or two and devel­oped an idea of what a horse was, that idea was the form of hors­e­ness now exist­ing in your mind. For Aris­totle, the rea­son that we can know things that our minds are capa­ble of intu­itively grasp­ing the forms of things we wit­ness in the world in our heads and then rea­son­ing about them. If forms did­n’t exist, we would­n’t be able to rea­son.

For many meta­phys­i­cal real­ists, that last point is key. The rea­son they embrace meta­phys­i­cal real­ism is that they main­tain that rea­son­ing and com­mu­ni­ca­tion are impos­si­ble if the con­cepts that we rea­son about are not in some sense real. If “red­ness” for exam­ple, is not some real prop­erty of dif­fer­ent things in the world, then we can’t mean­ing­fully use the word “red” to describe things. And, for meta­phys­i­cal real­ists, if red­ness is not some imma­te­r­ial form, dis­tinct from the red things them­selves, it can’t be a real prop­er­ty, pred­i­ca­ble of dif­fer­ent things in real­i­ty.


Now, meta­phys­i­cal real­ism has not gone unchal­lenged in phi­los­o­phy. There have been many philoso­phers over the cen­turies who have taken issue with the notion that imma­te­r­ial forms must be things that exist in order for human beings to be able to rea­son or com­mu­ni­cate. Per­haps the most famous chal­lenge to meta­phys­i­cal real­ism was that of nominalism in the late Mid­dle Ages, the notion that uni­ver­sals as such don’t exist and that words in gen­eral are just names that we apply to groups of par­tic­u­lar things for con­ve­nience. The most famous of the early nom­i­nal­ists was a late medieval scholas­tic philoso­pher who we call William of Ockham. Ock­ham was Fran­cis­can friar orig­i­nally from south­ern Eng­land famous for, among other things, his advo­cacy for the principle of simplicity1 as well as for his sup­port of nominalist meta­physics.

Ock­ham embraced nom­i­nal­ism because he thought that meta­phys­i­cal real­ism was self­-­con­tra­dic­to­ry. For Ock­ham, real­ism implies that a uni­ver­sal is both many things and a sin­gu­lar thing at the same time and that this was a con­tra­dic­tion. He argued that if a uni­ver­sal existed in the par­tic­u­lar things that instan­ti­ate it, then destroy­ing any par­tic­u­lar would destroy the uni­ver­sal it instan­ti­at­ed. Destroy­ing a horse for exam­ple, would destroy the uni­ver­sal “hors­e­ness” that every sin­gle horse in exis­tence par­tic­i­pated in. If that were not the case, then either “hors­e­ness” was not a sin­gle thing, or else it was not shared uni­ver­sally among all horses and in either case it would not be what philoso­phers call a uni­ver­sal.

So for Ock­ham, uni­ver­sals could­n’t be the thing par­tic­u­lar objects held in com­mon that let us talk about them. Instead, he argued for form of nom­i­nal­ism called conceptualism. He argued that we form concepts inside the mind, that cor­re­spond to groups of par­tic­u­lar objects that we expe­ri­ence in the world. For exam­ple, a per­son might encounter a num­ber of ani­mals that have long legs and which peo­ple ride upon. In see­ing that they are all pretty sim­i­lar, they might develop a con­cept in his mind for this sort of ani­mal and the name “horse” could then be applied to that con­cept. This con­cept is some­thing that exists only in the mind, and it func­tions to link together mul­ti­ple related expe­ri­ences. In this schema, words such as “red­ness” or “tri­an­gu­lar­i­ty” are not meta­phys­i­cal real­i­ties that exist inde­pen­dent of the mind which are then revealed to the mind via some meta­phys­i­cal or spir­i­tual mech­a­nism, but rather are con­structs built by the mind to unify sim­i­lar expe­ri­ences.

The Debate

Nom­i­nal­ism vs Meta­phys­i­cal Real­ism might seem like a bit of an eso­teric debate, but it was actu­ally quite a heated sub­ject in the late Mid­dle Ages and in the Early Mod­ern Peri­od. In fact, philoso­phers still debate the mer­its of either posi­tion. At the root of the debate is a con­cern about the objec­tiv­ity vs sub­jec­tiv­ity of lan­guage, log­ic, belief, even ethics and morals. Real­ist philoso­phers might argue that if “red­ness” for exam­ple were only a con­cept in the mind rather than a real prop­erty of things in real­i­ty, then the fact each indi­vid­ual might have his own pri­vate con­cept of “red­ness” might pre­clude any com­mu­ni­ca­tion about red­ness or red things, because if we don’t have the same con­cept of red­ness then we can’t be cer­tain that we are talk­ing about the same thing.

A Nom­i­nal­ist might object to that argu­ment, by point­ing out that “red­ness” is only a sen­sa­tion that occurs when light of a cer­tain wave­length hits the eyes and not a prop­erty of objects prop­er­ly. The sen­sa­tion of red­ness might be pro­duced in mul­ti­ple ways, so there isn’t a sin­gle uni­ver­sal prop­erty of objects that con­sti­tutes “red­ness”. And besides, dif­fer­ent peo­ple do have dif­fer­ent per­cep­tions of col­or, such as peo­ple who are red-­green col­or­blind and can’t dis­tin­guish “red­ness” from “green­ness” and peo­ple from other cul­tures who some­times dis­tin­guish col­ors dif­fer­ent­ly. A nom­i­nal­ist might con­tinue to point out that pop­u­lar debates such as “Is a hot­dog a sand­wich,” imply that dif­fer­ent peo­ple often have dif­fer­ent ideas about the cor­rect def­i­n­i­tion of a given word, that is, they have dif­fer­ent con­cepts that they use the same word for, and yet still man­age to mean­ing­fully com­mu­ni­cate with those words most of the time. This would imply that at least some uni­ver­sals, such as “sand­wich” are nom­i­nal­ist rather than real­ist in nature and that this has not caused the break­down in com­mu­ni­ca­tion that meta­phys­i­cal real­ists sug­gest would hap­pen. We may not all have the same con­cept of a sand­wich, but we can order them in restau­rants just fine. As the philoso­pher Karl Pop­per pointed out while defend­ing nominalism:

A term like “sand dune” or “wind” is cer­tainly very vague. How many inches high must a lit­tle sand hill be in order to be called “sand dune”? How quickly must the air move in order to be called “wind”? How­ev­er, for many of the geol­o­gists’ pur­pos­es, these terms are quite suf­fi­ciently pre­cise, and for other pur­pos­es, when a higher degree of dif­fer­en­ti­a­tion is need­ed, he could always say, “dunes between 4 and 30 feet high”, or “wind of a veloc­ity between 20 and 40 mile an hour”. And the posi­tion in the more exact sci­ences is anal­o­gous. In phys­i­cal mea­sure­ments, for instance, we always take care to con­sider the range within which there may be an error; and pre­ci­sion does not con­sist in try­ing to reduce this range to noth­ing, or in pre­tend­ing that there is no such range, but rather in its explicit recognition.2

A real­ist might not be sat­is­fied with this rebut­tal, argu­ing even if a nom­i­nal­ist approach can pro­vide a sat­is­fac­tory solu­tion to some uni­ver­sals, it still does­n’t apply to all. For exam­ple, a (Euclid­ean) tri­an­gle always has angles that sum up to 180°. We can deduce this with­out deal­ing with any tri­an­gu­lar object in par­tic­u­lar, merely by con­sid­er­ing the log­i­cal con­se­quences of “tri­an­gu­lar­i­ty”. From that it would seem that “tri­an­gu­lar­i­ty” is some­thing more than just a name used for objects that bear a cer­tain sim­i­lar­ity to each other and instead rep­re­sents some kind of meta­phys­i­cal real­i­ty, oth­er­wise we would­n’t be able to rea­son in abstrac­tion from par­tic­u­lar things.

Now, there are dif­fer­ent kinds of nom­i­nal­ist and dif­fer­ent kinds of meta­phys­i­cal real­ist and dif­fer­ent philoso­phers have dif­fer­ent takes as to what uni­ver­sals can be treated nom­i­nally and which kinds must be real. Some philoso­phers treat math­e­mat­i­cal con­cepts as real and oth­ers don’t. I think that it’s pos­si­ble to defend a nom­i­nal­ist inter­pre­ta­tion of math­e­mat­ics through reduc­tion­ism.

The Game of Life

In 1970 a math­e­mati­cian name John Con­way devised a sin­gle player game he called the Game of Life. This was a cel­lu­lar automa­ton mod­el, a form of sim­u­la­tion in which a grid of “cells” cycle through dif­fer­ent states based on the val­ues of their neigh­bor cells. These cel­lu­lar automata mod­els were invented to demon­strate a reduc­tion­ist model of self­-repli­ca­tion, that is with a cer­tain rule-set and ini­tial con­di­tion, a cer­tain pat­tern of cells could be made to repli­cate itself indef­i­nite­ly. Dif­fer­ent cel­lu­lar automata fol­lowed dif­fer­ent rules and had dif­fer­ent behav­ior, but Con­way’s Game of Life man­aged to be a par­tic­u­larly inter­est­ing vari­a­tion based on a rather sim­ple set of rules. In the Game of Life, cells are either “alive” or “dead” and between each gen­er­a­tion the state of the cells would change accord­ing to the rules as follows:

  1. Any live cell with fewer than two live neighbours dies, as if by underpopulation.
  2. Any live cell with two or three live neighbours lives on to the next generation.
  3. Any live cell with more than three live neighbours dies, as if by overpopulation.
  4. Any dead cell with exactly three live neighbours becomes a live cell, as if by reproduction.

The result was a sur­pris­ingly com­plex sys­tem with sur­pris­ingly com­plex attribut­es. One inter­est­ing attribute of the Game of Life is that there are many dif­fer­ent pat­terns that appear in the game. Some pat­terns appear nat­u­rally and fre­quently whereas oth­ers are uncom­mon or need to be cre­ated inten­tion­al­ly. Pat­terns have pre­dictable and some­time com­plex behav­ior. Some com­mon pat­terns are very sim­ple and have very sim­ple behav­ior, such as the “block” pat­tern which does­n’t change from one gen­er­a­tion or another or the “bea­con” pat­tern which appears to blink on and off. Some are a bit more com­plex, such as the “glider” which moves con­tin­u­ously in one direc­tion. In fact the behav­ior of pat­terns can be made arbi­trar­ily com­plex, with entire com­put­ers being sim­u­lated in a sin­gle game of life.

Many of the pat­terns in the Game of Life are recog­niz­able and appear fre­quent­ly. We have names for many pat­terns, includ­ing all the most com­mon ones. Yet these pat­terns, even the most com­plex ones, are all reducible to a sim­ple set of rules as listed above. There isn’t any such entity as a “glid­er” built into the rules of Life, but they nev­er­the­less appear spon­ta­neously as a result of many other pat­terns. In that sense, a nom­i­nal­ist account of these pat­terns makes more sense than a meta­phys­i­cal real­ist account. “Glid­er” is a name that we apply to a com­mon pat­tern that appears sim­i­lar to instances of a glid­er. The com­mon behav­ior is caused, not by a com­mon meta­phys­i­cal “glid­er­ness” shared by all glid­ers, but by the rules listed above. “Glid­ers” rather, are an emergent property of the Game of Life; we sim­ply invent the name “glid­er” to refer to them.

It’s worth point­ing out that most of the pat­terns that result in the Game of Life were not inten­tional and were not designed into the sys­tem. Con­way did not know what pat­terns would emerge when he set­tled on the rules listed above. Instead, they are the con­se­quences of the sys­tem as such, not of any par­tic­u­lar design on the part of the sys­tems mak­er.

We can anal­o­gously argue, that in the phys­i­cal world, geo­met­ric and other math­e­mat­i­cal con­structs such as tri­an­gles are them­selves emer­gent prop­er­ties of a phys­i­cal world ruled by a set of rules, includ­ing rules gov­ern­ing how mat­ter is arranged in a three­-di­men­sional space. It was not nec­es­sary that tri­an­gles have the prop­er­ties that we as­so­ciate with them. There are non-euclid­ean forms of geom­e­try in which the angles of a tri­an­gle don’t add up to 180°. And there are cer­tain math­e­mat­i­cal spaces, in which tri­an­gle inequal­ity does not hold. By this con­cep­tion, a tri­an­gle, like a glider or bea­con in the Game of Life, is an emer­gent prop­erty of space. Tri­an­gles are pos­si­ble because space is shaped in a cer­tain way, and tri­an­gles appear in nature because var­i­ous nat­ural processes result in them. That tri­an­gles seem to have com­mon prop­er­ties is due not to a uni­ver­sal “tri­an­gu­lar­i­ty” that all tri­an­gles have in com­mon, but to the fact that tri­an­gles are a result of the way space is shaped. The shape of space is what tri­an­gles have in com­mon, and we can rea­son about tri­an­gles because we can rea­son about objects in space.

A die-hard meta­phys­i­cal real­ist might insist that the shape of space can instead con­sti­tute a uni­ver­sal and have other uni­ver­sals be reducible to that com­mon base uni­ver­sal. How­ev­er, tra­di­tional nom­i­nal­ists such as Ock­ham did­n’t object to the idea of objects exist­ing in a com­mon space or being sub­ject to com­mon rules; they objected to con­cepts like “horse” hav­ing a dis­tinct real­ity inde­pen­dent of the objects we call “hors­es”. A sit­u­a­tion where horses are reducible to phys­i­cal laws still allows for a nom­i­nal­ist approach to hors­es. We have an under­stand­ing of what horses are because we expe­ri­ence them indi­vid­u­ally in real­i­ty, not because we directly per­ceive the fun­da­men­tal laws of nature and how they can result in hors­es. So for any prac­ti­cal pur­pose, horses are under­stood in a nom­i­nal­ist fash­ion, as a word for a group of things that exist in nature that seem sim­i­lar, rather than in the real­ist man­ner of directly per­ceiv­ing the essence of hors­e­ness when­ever we encounter a horse.


The inter­est­ing thing about this approach is that it flips the whole debate on its head. Tra­di­tional scholas­tic phi­los­o­phy con­cep­tu­al­izes the phys­i­cal world as mat­ter shaped by forms, these forms essen­tially being uni­ver­sal essences and our abil­ity to engage with and under­stand the world around us is because of the struc­ture pro­vided by those forms. A more con­tem­po­rary under­stand­ing instead con­cep­tu­al­izes the phys­i­cal world as com­posed of mat­ter which fol­lows a fixed set of rules. In this way, con­sis­tency and the result­ing intel­li­gi­bil­ity result from the fact that objects are part of the world which is intel­li­gi­ble rather than the world being intel­li­gi­ble because it is pop­u­lated by intel­li­gi­ble objects. The world is intel­li­gi­ble because of the con­sis­tency of the rules which gov­ern it, a pre­sup­po­si­tion we call uniformitarianism. That is the begin­ning of mod­ern sci­ence.

  1. This is sometimes called “Ockham’s Razor” after him. Ockham himself, however, never used the word “razor” to describe the principle of simplicity. 
  2. K. Popper - The Open Society and its Enemies 

Last update: 19/01/2023

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