# Preamble

I’ve wanted to write a se­ries of posts here on logic and the foun­da­tions of math­e­mat­ics for a while now. There’s been some re­cent dis­cus­sion about the on­tol­ogy of num­bers and the ex­is­tence of math­e­mat­i­cal en­tities, so this seems as good a time as any to start.

Many of the dis­cussed philo­soph­i­cal prob­lems, as far as I can tell, stem from the as­sump­tion of for­mal­ism. That is, many peo­ple seem to think that math­e­mat­ics is, at some level, a for­mal logic, or at least that the ac­tivity of math­e­mat­ics has to be founded on some for­mal logic, es­pe­cially a clas­si­cal one. Beyond that be­ing an un­ten­able po­si­tion since Gödel’s In­com­plete­ness The­o­rems, it also doesn’t make a whole lot of in­tu­itive sense since math­e­mat­ics was clearly done be­fore the in­ven­tion of for­mal logic. By aban­don­ing this as­sump­tion, and tak­ing a more con­struc­tivist ap­proach, we get a much clearer view of math­e­mat­ics and logic as a whole.

This first post is mostly in­for­mal philoso­phiz­ing, at­tempt­ing to de­scribe ex­actly what logic and math­e­mat­ics is about. My sec­ond post will be a more tech­ni­cal dis­cus­sion ac­count­ing for the ba­sic no­tions of logic.

# In­tu­itions and Sensations

To be­gin, I’d like to point out a fact which most would find ob­vi­ous but has, in the past, lead to difficult philo­soph­i­cal prob­lems. It is clear that we don’t have di­rect ac­cess to the real world. In­stead, we have senses which feed in­for­ma­tion, even if dishon­estly, to our mind. Th­ese senses may be pre­dictable, may be po­ten­tially mod­eled by a pat­tern which mimics our stream of senses. At some level, we have di­rect ac­cess to a sen­sory sig­nal. This sig­nal is not a pure, un­filtered lens on the world, but it is a sig­nal, in­de­pen­dent of, but di­rectly ac­cessible by, us.

We also have ac­cess to our in­tu­itions, the part of our thoughts which we may la­bel “ideas”. We may not have to­tal ac­cess to all our fac­ul­ties. Much of our men­tal pro­cess­ing is done out­side the view of our aware­ness. If I asked you to name a ran­dom city, even­tu­ally you’d come up with one. You’d, how­ever, be hard-pressed to pro­duce a trace of how that city’s name came to your aware­ness. Per­haps you could offer back­ground in­for­ma­tion, ex­plain­ing why you’d name that city, among all the pos­si­bil­ities. Re­gard­less of such ac­counts, we’d still lack a trace of the sig­nal sent by your con­scious­ness (“I want the name of a city, any city”) reach­ing into the part of your mind ca­pa­ble of fulfilling such a re­quest, and the sub­se­quent re­cep­tion of the name into your aware­ness. We don’t have such de­tailed ac­cess to the in­ner-work­ings of our mind.

It seems that those things which we have di­rect ac­cess to are, in fact, part of us. Those in­tu­itions within our aware­ness, those filtered sig­nals which we di­rectly ex­pe­rience, make up our qualia, are in­stan­ti­ated in the sub­strate of our con­scious­ness. They may be thought of as part of our­selves, and to say we have ac­cess to them is to say that we have di­rect ac­cess to those parts of our­selves of which we are aware. This, I think, is triv­ially true. Though that isn’t es­sen­tial for the rest of this piece.

We may dis­t­in­guish nor­mal in­tu­itions from senses by the de­gree we can con­trol them. In­tu­itions are con­trol­lable and ma­nipu­la­ble by our­selves, while senses are not. This isn’t perfectly clean. One may, for ex­am­ple through small DMT doses, cause one to ex­pe­rience con­trol­lable hal­lu­ci­na­tions which are a man­i­fes­ta­tion of di­rect (though not com­plete) con­trol of the senses. Also, there are plenty of ex­am­ples of in­tu­itions which we find difficult to con­trol, such as ear-worms. For the sake of this work, I will ig­nore such cases. What I want to fo­cus on are sen­sory sen­sa­tions fed to our aware­ness pas­sively and those in­tu­itions which we have com­plete (or com­plete for prac­ti­cal pur­poses) con­trol. Th­ese are the sorts of things needed for logic, math­e­mat­ics, and sci­ence, which will be the pri­mary fo­cuses of this se­ries. For the re­main­der, by “sense” and “sen­sory data” I am refer­ring to those qualia which are ex­pe­rienced pas­sively, with­out de­liber­ate con­trol; by “in­tu­ition” I am refer­ring to those in­tu­itions which are un­der our di­rect and (at least ap­par­ent) to­tal con­trol.

# Grounding

At this stage, it’s use­ful to make a re­mark about lan­guage and ground­ing. Con­sider what I might be say­ing if I de­scribe some­thing as an elephant. Within my mind is an in­tu­ition which I’m as­sign­ing the word “elephant”, and in call­ing some­thing pre­sumed ex­ter­nal to me an elephant, I am as­sert­ing that my in­tu­ition is an ap­prox­i­mate model for the thing I’m nam­ing. The differ­ence be­tween the in­tu­ition and the real thing is im­por­tant. It is prac­ti­cally im­pos­si­ble to have a perfect un­der­stand­ing of real-world en­tities. My in­tu­ition tied to “elephant” does not con­tain all that I might con­sider know­able about elephants, but only those things which I do know. A vet­eri­nar­ian spe­cial­iz­ing in elephants would cer­tainly have a more ac­cu­rate, more elab­o­rate in­tu­ition as­signed to “elephant” than a non-spe­cial­ist, and this wouldn’t be the full ex­tent to which elephants could be mod­eled. In essence, I’m us­ing this mod­el­ing in­tu­ition as a metaphor for an elephant when­ever I use that word.

Based on this, we can ac­count for learn­ing and dis­agree­ment. Learn­ing can be char­ac­ter­ized as the pro­cess of re­fin­ing an ap­prox­i­mately cor­rect in­tu­ition mod­el­ing some­thing ex­ter­nal. A dis­agree­ment stems from two main places. Firstly, two peo­ple with similar sen­sa­tions may be us­ing differ­ing mod­els. From these differ­ences, two peo­ple may de­scribe iden­ti­cal sen­sa­tions differ­ently, as their mod­els might dis­agree. Se­condly, two peo­ple may think they’re get­ting similar sen­sa­tions when they are not, and so dis­agree be­cause they are un­able to cor­rectly com­pare mod­els, to be­gin with. This is the “Blind men and an elephant” sce­nario.

This ac­count also cleanly ex­plains why we can still mean­ingfully talk about elephants when none are pre­sent. In that case, we are speak­ing of the in­tu­ition as­signed to “elephant”. Ad­di­tion­ally, we can talk about non-ex­is­tent en­tities like uni­corns un­prob­le­mat­i­cally, as such things would still have re­al­ities as in­tu­itions. An as­ser­tion of ex­is­tence or nonex­is­tence is re­ally about an in­tu­ition, a model of some­thing. The prop­erty of ex­is­tence cor­re­sponds to a pre­dic­tion of pres­ence in the real world by our model, non-ex­is­tence to our model pre­dict­ing ab­sence. The cor­rect­ness of these prop­er­ties is pre­cisely the de­gree to which they ac­cu­rately pre­dict sen­sory data.

In­tu­itions need not be de­signed to model some­thing in or­der for it to be used to model some­thing else. If I try to de­scribe an an­i­mal which I’m only the first time en­coun­ter­ing, then I may con­struct a new model of it by piec­ing to­gether parts of older mod­els. I may even call it an “elephant-like-thing” if I feel it has some, if limited, pre­dic­tive power. In this way, I’m con­struct­ing a new model by char­ac­ter­iz­ing the de­gree to which other mod­els pre­dict prop­er­ties of the new an­i­mal I’m see­ing. Even­tu­ally, I may as­sign this new model a word, or bor­row a word from some­one else.

One can also cre­ate in­tu­itions with­out at­tempt­ing to model some­thing ex­ter­nal. If you were a mind in a void, with­out any sen­sory in­for­ma­tion, you should still be able to think of ba­sic math­e­mat­i­cal and log­i­cal con­cepts, such as num­bers. You might not be mo­ti­vated to do so, but the abil­ity to do so is what’s rele­vant here. Th­ese con­cepts can be un­der­stood in to­tal­ity as in­tu­itions, com­pletely defin­able with­out ex­ter­nal refer­ents. Later, this will be elab­o­rated on at length, but take this para­graph as-is for the mo­ment.

Even if an in­tu­ition was cre­ated with­out in­tent to model, it still can be used as such. For ex­am­ple, one can think of “2” with­out us­ing it to model any­thing. One can still say that a herd of elephants has 2 mem­bers, us­ing the in­tu­ition of 2 as a metaphor for some as­pect of the herd.

Some no­tion I’ve heard be­fore is that it seems like a herd with 2 mem­bers would have 2 mem­bers even if there was no one around to think so, and so 2 has to ex­ist in­de­pen­dently of a mind. Un­der my ac­count, this state­ment fails to un­der­stand per­spec­tive. It is cer­tainly the case that one could model a herd of 2 us­ing 2, re­gard­less of if any­one else was think­ing of the herd. How­ever, even ask­ing about the herd pre­sup­poses that at least the asker is think­ing about the herd, dis­prov­ing the premise that the herd isn’t be­ing thought about. If it were truly the case that no one was think­ing of it at all, then there’s noth­ing to talk about. The ques­tion would not have been asked in the first place, and the ap­par­ent prob­lem then van­ishes. It is clear at this point that stat­ing “a herd has 2 mem­bers” does not make 2 part of our model of the world.

At this point, I will in­tro­duce ter­minol­ogy which dis­t­in­guishes be­tween the two kinds of in­tu­itions dis­cussed. In­tu­itions which are po­ten­tially in­com­plete, de­signed to model ex­ter­nal en­tities will be called grounded in­tu­itions. Those in­tu­itions which may be com­plete and may ex­ist with­out mod­el­ing prop­er­ties will sim­ply be un­grounded in­tu­itions.

One com­mon de­scrip­tion of re­al­ity stem­ming from Pla­ton­ism is that of an im­perfect shadow or re­flec­tion of the tran­scen­den­tal world of ideals. After all, cir­cles are perfect, but noth­ing in the world de­scribed as a cir­cle is a truly perfect cir­cle. By my ac­count, perfec­tion doesn’t come into the pic­ture. A cir­cle is an un­grounded in­tu­ition. An ex­ter­nal en­tity is only ac­cu­rately called a cir­cle in so far as the in­tu­ition of a cir­cle ac­cu­rately mod­els the en­tity’s phys­i­cal form. The en­tity isn’t im­perfect, in some ob­jec­tive sense. Rather, the grounded in­tu­ition of that en­tity is sim­ply more com­plex than the un­grounded in­tu­ition of the cir­cle. The ap­par­ent im­perfec­tion of the world is only a man­i­fes­ta­tion of its com­plex­ity. Grounded in­tu­itions tend to be more com­pli­cated than the un­grounded in­tu­itions which we used to ap­prox­i­mate the real world. This is, at once, not sur­pris­ing, but sig­nifi­cant. If we lived in an ex­tremely sim­ple world (or one which was sim­ple rel­a­tive to our minds) then we might cre­ate un­grounded in­tu­itions which were sim­pler than the av­er­age un­grounded one. We may then have trou­ble dis­t­in­guish­ing be­tween sen­sory data and in­tu­ition, as all facts about the real world would be com­pletely ob­vi­ous and in­tu­itively pre­dictable.

# On­tolog­i­cal Com­mit­ments of Un­grounded Entities

I think it’s worth tak­ing the time to dis­cuss some con­tent re­lated to on­tolog­i­cal com­mit­ments and con­ven­tions. On­tolog­i­cal com­mit­ments were in­tro­duced by Quine, but I won’t hold true to the no­tion as he origi­nally de­scribed it. In­stead, by an on­tolog­i­cal com­mit­ment, I am refer­ring to an as­ser­tion of the ob­jec­tive ex­is­tence of an en­tity which is in­de­pen­dent of the sub­jec­tive ex­pe­rience of the per­son mak­ing the as­ser­tion.

Let’s take a sce­nario where two peo­ple are ar­gu­ing over what color the blood of a uni­corn is. One says silver, the other red. Our goal is to make sense of this ar­gu­ment. As­sum­ing nei­ther peo­ple be­lieve uni­corns ex­ist, what con­tent does this ar­gu­ment ac­tu­ally have?

First, it be­hooves us to make sense of what a uni­corn is, and what com­mit­ments we make in talk­ing about them. For the mo­ment, I’ll stick to a con­ven­tional dis­tribu­tive-se­man­ti­cal char­ac­ter­i­za­tion of mean­ing (I plan on mak­ing a post about this quite some time from now). Through our ex­pe­rience, we even­tu­ally as­so­ci­ate words like “blood”, “horse”, and “horn” with vec­tors in­side of some se­man­tic space. We can then com­bine them in a sen­si­cal way to pro­duce the idea of a horse with a horn, a new vec­tor for a new idea, a uni­corn. When talk­ing about com­mit­ments, we need to make a dis­tinc­tion be­tween two things; com­mit­ments to ex­pec­ta­tions, and com­mit­ments to ideas. When we define uni­corns in this man­ner, we are com­mit­ting our­selves to the idea of uni­corns as some­thing that’s co­her­ent and leg­ible. We are not mak­ing a com­mit­ment to uni­corns ex­ist­ing for real, that is we do not sud­denly ex­pect to see a uni­corn in real life. This may be con­sid­ered an on­tolog­i­cal com­mit­ment of a sort. We cer­tainly as­cribe ex­is­tence to the idea of a uni­corn, at least within our own mind. We don’t, how­ever, on­tolog­i­cally com­mit our­selves to what the idea of uni­corns might the­o­ret­i­cally model. Since all sen­tences can­not help but re­fer to ideas rather than ac­tual en­tities, re­gard­less of our ex­pec­ta­tions, the as­ser­tion that uni­corn blood is silver per­tains to this idea of uni­corns, noth­ing that ex­ists out­side of our mind.

If a tree falls in a for­est and no one is around to hear it, does it make a sound?

This ques­tion has a stan­dard solu­tion that I’d con­sider uni­ver­sally satis­fac­tory. Ul­ti­mately, the ques­tion isn’t about re­al­ity, it’s about the defi­ni­tion of the word “sound”. If by “sound” the asker is speak­ing of a sen­sa­tion in the ear, then the an­swer is “no”. If they mean vibra­tions in the air, then the an­swer is “yes”. Un­der the dis­tri­bu­tional se­man­tics of the word “sound”, we can talk about this word hav­ing val­ues in var­i­ous di­rec­tions. For some peo­ple, “sound” is as­signed the re­gion defined by a pos­i­tive value in the di­rec­tion cor­re­spond­ing to sen­sa­tions in the ear. For oth­ers, “sound” is as­signed to the re­gion with pos­i­tive value in the di­rec­tion cor­re­spond­ing to vibra­tions in the air. Th­ese two re­gions have heavy over­lap in prac­tice. When we ex­pe­rience a sen­sa­tion, it’s rare for it to have a pos­i­tive value in one of these, but not the other. And so, we as­sign one of these re­gions the word “sound”, most of the time hav­ing no prob­lem with oth­ers who make a differ­ent choice but ar­riv­ing at dis­agree­ments over ques­tions like the above.

But which is it? What does “sound” ac­tu­ally mean? Well, that’s a choice. Con­sider the situ­a­tion in de­tail. Is there any­thing that needs to be clar­ified? Are there vibra­tions in the air? Yes. Are there any sen­sa­tions in an ear caused by these vibra­tions? No. So there’s noth­ing left to learn. All that’s left is to de­cide how to de­scribe re­al­ity. It may even be use­ful to split the term, to talk about “type-1 sound” and “type-2 sound”, which usu­ally co­in­cide, but don’t on rare oc­ca­sions. Re­gard­less, it’s a mat­ter of con­ven­tion, not a mat­ter of fact, whether the word “sound” should ap­ply.

And so, we’re in sight of the re­s­olu­tion to the uni­corn blood ar­gu­ment. One per­son has a re­gion in their se­man­tic space cor­re­spond­ing to one-horned horses with silver blood, and want’s to as­sign that re­gion the word “uni­corn”. The other per­son has iden­ti­fied a close-by se­man­tic re­gion, but there the blood is red, and they want that to have the word “uni­corn”. Note that nei­ther would think that the oth­ers claim is non­sense. The ar­gu­ment is not pred­i­cated on, for ex­am­ple, one per­son think­ing the idea of a uni­corn with red blood is in­co­her­ent. Both par­ties agree that each other have iden­ti­fied mean­ingful re­gions of se­man­tic space. They are mak­ing iden­ti­cal on­tolog­i­cal com­mit­ments. What they are dis­agree­ing on is a nam­ing con­ven­tion.

Through­out this se­ries, I will of­ten dis­cuss math­e­mat­ics and logic as fun­da­men­tally sub­jec­tive ac­tivi­ties, but this does not mean I re­ject math­e­mat­i­cal ob­jec­tivism as such. Rather, the ob­jec­tive char­ac­ter of math­e­mat­ics moves from be­ing an as­pect of math­e­mat­ics it­self to be­ing an as­pect of how it’s prac­ticed. Math­e­mat­ics is done as a so­cial ac­tivity car­ried by a con­ven­tion which is it­self ob­jec­tive: or at least (ideally) as ob­jec­tive as a ruler. Show­ing that some­one is math­e­mat­i­cally wrong largely boils down to show­ing which con­ven­tion a per­son is break­ing in mak­ing an in­cor­rect judg­ment.

Brouwer, who was the first to re­ally push math­e­mat­i­cal in­tu­ition­ism, de­scribed math­e­mat­ics as a so­cial ac­tivity at its core. As a con­se­quence, he ar­gued against the idea of a for­mal log­i­cal foun­da­tion be­fore Gödel’s in­com­plete­ness the­o­rems were even dis­cov­ered.

The ba­sic idea of con­struc­tivism is to limit our on­tolog­i­cal com­mit­ments as much as pos­si­ble. Con­sider the well known “I think, there­fore I am”. It high­lights the fact that the act of think­ing and in­tro­spec­tion it­self im­plies an on­tolog­i­cal com­mit­ment to the self. Since we are already do­ing those things, it’s re­ally not much of a com­mit­ment at all. Similarly, the fact that I am writ­ing in a lan­guage com­mits me on­tolog­i­cally to the ex­is­tence of the lan­guage I’m writ­ing in. As I’m do­ing this any­way, it’s not much of a com­mit­ment. For this, I call these sorts of com­mit­ments “cheap com­mit­ments”.

Math­e­mat­i­cal and log­i­cal en­tities are ideas. By dis­cussing them, we are com­mit­ting our­selves to the ex­is­tence of these en­tities at least as ideas. For ex­am­ple, if I say “there ex­ists an even nat­u­ral num­ber”, I am com­mit­ting my­self to the ideas of nat­u­ral num­bers and even­ness. I’m also com­mit­ting my­self to the co­her­ence or sound­ness of these ideas, that the state­ment in ques­tion is mean­ingful mod­ulo the se­man­tics of the ideas used.

I can eas­ily make gram­mat­i­cal-look­ing sen­tences that seem to make some sort of ex­pen­sive com­mit­ment. For ex­am­ple, I could say that g’gle­mors ex­ist and that a h’plop is an ex­am­ple of a g’gle­mor on ac­count of hipl’xtheth. If I said those things with any sort of se­ri­ous­ness I’d be com­mit­ting my­self to the ex­is­tence of those men­tioned things at least as ideas, as well as the sound­ness of those ideas. Be­ing non­sense words not rep­re­sent­ing any­thing at all, I’d ob­vi­ously be mis­guided in mak­ing such com­mit­ments, they cer­tainly aren’t cheap.

The point of a con­struc­tivist ac­count is to de­scribe math­e­mat­i­cal and log­i­cal ideas in such a way that one is com­mit­ted to their sound­ness in a cheap way. And here we can start to see the sig­nifi­cance of char­ac­ter­iz­ing math­e­mat­ics and logic as be­ing about un­grounded en­tities. In or­der for my com­mit­ments to those ideas to be cheap, they must be to­tally char­ac­ter­ized by some­thing that comes from within me, by some­thing that I’m do­ing any­way when dis­cussing those ideas.

# Precom­mit­ments and Judgments

We say that an idea is a cheap com­mit­ment if, in defin­ing the no­tion, we sum­mon the en­tity be­ing defined, or perform the ac­tivity which we are judg­ing to be the case. In or­der to do this, we need to pay at­ten­tion to pre­com­mit­ments.

A pre­com­mit­ment is a pre­scrip­tion we make of our own be­hav­ior. It’s an ac­tivity which is be­ing done so long as those pre­scrip­tions are be­ing fol­lowed. Precom­mit­ments are the core of struc­tured think­ing. When­ever we im­pose any pat­tern or con­sis­tency to our think­ing, we are mak­ing a pre­com­mit­ment. By an­a­lyz­ing our pre­com­mit­ments closely, we can con­struct, ex­plic­itly, ideas which are cheap on­tolog­i­cal com­mit­ments. If we are ac­tively do­ing a pre­com­mit­ment, then we can cheaply ac­knowl­edge the ex­is­tence of the idea con­jured by this pre­com­mit­ment.

Many un­grounded in­tu­itions arise as a form of mean­ing-as-us­age. Some words don’t have mean­ing be­yond the pre­cise way they are used. If you take a word like “elephant”, it’s mean­ing is con­tin­gent on ex­ter­nal in­for­ma­tion which may change over time. A word like “and”, how­ever, isn’t. As a re­sult, we’d say “and”’s mean­ing fun­da­men­tally boils down to how it’s used, and noth­ing more. Go­ing be­yond that, if we are to fo­cus on un­grounded in­tu­itions which are com­plete and com­pre­hen­si­ble, then we are fo­cus­ing pre­cisely on those un­grounded in­tu­itions who’s defi­ni­tion is pre­cisely a speci­fi­ca­tion of us­age, and noth­ing more. That speci­fi­ca­tion of us­age is our pre­com­mit­ment. Of course, us­age hap­pens out­side the mind, but the rules dic­tat­ing that us­age aren’t, and its those canon­i­cal rules of us­age which I mean by “defi­ni­tion”.

The ba­sic el­e­ments of defi­ni­tions are judg­ments. Judg­ments in­clude things like judg­ing that some­thing is a propo­si­tion, or is a pro­gram, or is some other syn­tac­tic con­struc­tion. Judg­ments also in­clude as­ser­tions of truth, false­hood, pos­si­bil­ity, val­idity, etc of some data. How­ever, be aware that a judg­ment sim­ply con­sists of a pat­tern of men­tal to­kens which we may de­clare. Re­gard­less of what pre­con­cep­tions about pos­si­bil­ity, truth, etc. one has, these should be over­writ­ten by the com­pleted mean­ing ex­pla­na­tion in or­der to be un­der­stood as a purely un­grounded in­tu­ition and a cheap com­mit­ment.

When we make a judg­ment, we are merely as­sert­ing that we may use that pat­tern in our rea­son­ing. Precom­mit­ments, as we will make use of them here, are a col­lec­tion of judg­ments. As a con­se­quence, what we are pre­com­mit­ting our­selves to is an al­lowance of us­age for cer­tain pat­terns of men­tal to­kens when rea­son­ing about a con­cept. The full pre­com­mit­ment sum­mon­ing some con­cept will be called the mean­ing ex­pla­na­tion for that con­cept.

Ul­ti­mately, it is ei­ther the case that we make a par­tic­u­lar judg­ment or we don’t. That, how­ever, is a fact about our own be­hav­ior, not about the na­ture of re­al­ity in to­tal, in essence. Fur­ther­more, some­one not mak­ing a par­tic­u­lar judg­ment is not au­to­mat­i­cally mak­ing the op­po­site, or negated, judg­ment. In fact, such a thing doesn’t even make sense in gen­eral. As a re­sult, we don’t re­pro­duce clas­si­cal logic. Though, as we’ll even­tu­ally see, there are con­struc­tive log­ics which are clas­si­cal. How­ever, it’s worth dis­pel­ling the idea that there’s “one true logic”. Ques­tions about which kind of log­i­cal sym­bols, clas­si­cal, in­tu­ition­is­tic, lin­ear, etc. is the “true” one are non­sense. One is only cor­rect rel­a­tive to some prob­lem which has an el­e­ment which is to be mod­eled by one of these. Whichever is the more ac­cu­rate model is the cor­rect one, there is no “one true logic”, and it’s cer­tainly not the case that the in­tu­itions which make up math­e­mat­ics are gov­erned by a clas­si­cal logic. For ex­am­ple, the ex­is­tence of the­o­ret­i­cally un­solv­able prob­lems (e.g. the halt­ing prob­lem) illus­trates that our ca­pac­ity for judg­ing truth is fun­da­men­tally con­strained, not by some ob­jec­tive tran­scen­den­tal stan­dard for truth, but rather by our abil­ity to make proofs.

To sum­ma­rize, to define a con­cept we give a list of judg­ments, rules dic­tat­ing which pat­terns of to­kens we can use when con­sid­er­ing the con­cept. So long as these rules are be­ing fol­lowed, the con­cept ex­ists as a co­her­ent idea. If the pre­com­mit­ment is vi­o­lated, for ex­am­ple by mak­ing a judg­ment about the con­cept which is not pre­scribed by the rules, then the con­cept, as defined by the origi­nal pre­com­mit­ment, no longer ex­ists. There may be a new pre­com­mit­ment that defines a differ­ent con­cept us­ing the same to­kens which is not vi­o­lated, but that, be­ing a differ­ent pre­com­mit­ment, con­sti­tutes a differ­ent mean­ing ex­pla­na­tion, and so its sum­moned con­cept does not have the same mean­ing. So long as I fol­low a pre­com­mit­ment defin­ing a con­cept, it is hyp­o­crit­i­cal of me to deny the co­her­ence of that con­cept, just as it would be hyp­o­crit­i­cal to deny my lan­guage as I speak, to deny my ex­is­tence so long as I live.

# Com­pu­ta­tion to Canon­i­cal Form

We are now free to ex­plore an ex­am­ple of the con­struc­tion of an un­grounded in­tu­ition. I should be spe­cific and point out that not all un­grounded in­tu­itions are un­der dis­cus­sion. For the sake of math­e­mat­ics and logic, in­tu­itions must be com­pletely com­pre­hen­si­ble. Un­like grounded in­tu­itions, an un­grounded one may be such that it’s never mod­ified by new in­for­ma­tion. This doesn’t de­scribe all un­grounded in­tu­itions, but it de­scribes the ones we’re in­ter­ested in.

One of the most im­por­tant judg­ments we will con­sider is of the form . It is a kind of com­pu­ta­tional judg­ment. It’s worth ex­plain­ing why com­pu­ta­tion is con­sid­ered be­fore any­thing else in math­e­mat­ics. To digress a bit, it’s easy to ar­gue that some no­tion of com­pu­ta­tion is nec­es­sary for do­ing even the most ba­sic as­pects of or­di­nary math­e­mat­ics. Con­sider, for ex­am­ple, the stan­dard the­o­rem; for all propo­si­tions and , . The uni­ver­sal quan­tifi­ca­tion al­lows us to perform a sub­sti­tu­tion, get­ting, for ex­am­ple, , as an in­stance.

We should med­i­tate on sub­sti­tu­tion, an es­sen­tial re­quire­ment of even the most ba­sic and an­cient as­pects of logic. Sub­sti­tu­tion is an al­gorithm, a com­pu­ta­tion which must be performed some­how. In or­der to re­al­ize , we must be do­ing the ac­tivity cor­re­spond­ing to the sub­sti­tu­tion of with and the ac­tion cor­re­spond­ing to the sub­sti­tu­tion of with at some point. Sub­sti­tu­tion will ap­pear over and over again in var­i­ous guises, act­ing as a cen­tral and pow­er­ful no­tion of com­pu­ta­tion. To em­pha­size, once sub­sti­tu­tion is available, we are of the way to­ward com­plete and fully gen­eral Tur­ing-Com­plete com­pu­ta­tion via the lambda calcu­lus. Much of the miss­ing fea­tures per­tain to ex­plicit vari­able bind­ing, which we need any­way in or­der to use the quan­tifiers of first-or­der logic. I don’t think it’s re­ally de­bat­able that com­pu­ta­tion on­tolog­i­cally pre­cedes logic. One can do logic as an ac­tivity, and much of that ac­tivity is com­pu­ta­tional in na­ture.

Be­fore ex­posit­ing on some ex­am­ple judg­ments, we should ad­dress the need for iso­lat­ing con­cepts. Con­sider a the­ory with nat­u­ral num­bers and prod­ucts . We must ask what con­sti­tutes a nat­u­ral num­ber and a product. By de­fault, we can form a nat­u­ral num­ber as ei­ther zero or the suc­ces­sor of a nat­u­ral num­ber. e.g. , , , , … A product can be formed via where is an and is a . Ad­di­tion­ally, we have that, if is a nat­u­ral num­ber then (where is a pro­jec­tion func­tion) is a nat­u­ral num­ber, and if is a nat­u­ral num­ber then is a nat­u­ral num­ber, and if is a nat­u­ral num­ber then is a nat­u­ral num­ber, etc. to in­finity. This situ­a­tion gets branch­ingly more com­plex as we add new con­cepts to our the­ory. If we don’t define con­cepts as fun­da­men­tally iso­lated from each other, we in­hibit the ex­ten­si­bil­ity of our logic. This is both un­prag­matic and un­re­al­is­tic, as we will want to ex­tend the breadth of con­cepts we can deal with as we model more novel things. Fur­ther­more, the co­her­ence of the con­cept of a nat­u­ral num­ber should not de­pend on the co­her­ence of the no­tion of a product. Ul­ti­mately, each con­cept should be defined by some pre­com­mit­ment con­sist­ing of a list of rules for mak­ing judg­ments. If we en­ter­tain this in­finite regress, then there may be no way in gen­eral to state what the pre­com­mit­ment in ques­tion even is.

At the core of our defi­ni­tions will be canon­i­cal forms. Every time we define a new con­cept, we will as­sert what its canon­i­cal forms are. For ex­am­ple, in defin­ing the nat­u­ral num­bers we will judge that and that, as­sum­ing , we can con­clude that . We can’t as­sume this alone, how­ever. Con­sider, for ex­am­ple , which should be a nat­u­ral num­ber, but isn’t in the cor­rect form. We now have an op­por­tu­nity to ex­plain . in­di­cates that we start out with some men­tal in­stan­ti­a­tion , and af­ter some men­tal at­ten­tion, it be­comes the in­stan­ti­a­tion . So we have, for ex­am­ple . When I say , I do not mean that is equal to . That’s a sep­a­rate kind of judg­ment. This means our full judg­ment is that iff or for some . There are some de­tails miss­ing from this defi­ni­tion, but it should serve as a guid­ing ex­am­ple, the first rough sketch of what I mean by a mean­ing ex­pla­na­tion.

It is worth di­gress­ing some­what to cri­tique the ax­io­matic method. Most peo­ple, es­pe­cially when first learn­ing of a sub­ject, will ex­pe­rience a math­e­mat­i­cal or log­i­cal con­cept as a grounded in­tu­ition. This is re­flected in a per­son’s an­swer to ques­tions such as “why is ad­di­tion com­mu­ta­tive?”. Most peo­ple could not an­swer. It is not part of the defi­ni­tion of ad­di­tion or num­bers for this prop­erty to hold. Rather, this is a prop­erty stem­ming from more so­phis­ti­cated rea­son­ing in­volv­ing math­e­mat­i­cal in­duc­tion. A per­son can, none the less, feel an un­der­stand­ing of math­e­mat­i­cal con­cepts and an ac­cep­tance of prop­er­ties of them with­out knowl­edge of their un­der­ly­ing defi­ni­tions. Ax­io­matic meth­ods, such as the ax­ioms of ZFC, don’t ac­tu­ally define what they are about. In­stead, they list prop­er­ties that their topic must satisfy.

The no­tion of ZFC-set, in some sense, is grounded by an un­der­stand­ing of the ax­ioms, though it is still tech­ni­cally an un­grounded in­tu­ition. This state of af­fairs holds for any ax­io­matic sys­tem. There is some­thing fun­da­men­tally un­grounded about a for­mal logic, but it’s not the con­cepts which the ax­ioms de­scribe. Rather, what we have in a for­mal logic is a mean­ing ex­pla­na­tion for the logic it­self. That is, the ax­ioms of the logic tell us pre­cisely what con­sti­tutes a proof in the logic. In this way, we may for­mu­late a mean­ing ex­pla­na­tion for any for­mal logic, con­sist­ing of judg­ments for each ax­iom and rule of in­fer­ence. Con­se­quently, we can cheaply com­mit our­selves to the co­her­ence of the logic as an idea. What we can’t cheaply com­mit our­selves to are the ideas ex­pressed within the logic. After all, a for­mal logic could be in­con­sis­tent, it’s ideas may be in­co­her­ent.

As a con­se­quence, the no­tion of a co­her­ent idea of ZFC-set can­not be com­mit­ted to cheaply. This holds similarly for any con­cept de­scribed purely in terms of ax­ioms. It might be made cheap by ap­peal­ing to a suffi­cient mean­ing ex­pla­na­tion, but with­out ad­di­tional effort, things treated purely ax­io­mat­i­cally lack proper defi­ni­tions in the sense used here.

• Really ap­pre­ci­ate this post as it takes what I view to be a suffi­ciently skep­ti­cal, naive, be­gin­ner’s philo­soph­i­cal view. I look for­ward to the fol­lowups. Also, if Medium weren’t so bad at dis­play­ing math, I’d ask you if you’d be in­ter­ested in pub­lish­ing this se­ries over on Map and Ter­ri­tory.

• How do Gödel’s in­com­plete­ness the­o­rems make it “un­ten­able” to think that math­e­mat­ics is “founded on some for­mal logic”?

They show that if you do math­e­mat­ics that way and don’t make mis­takes then some ques­tions will re­main unan­swer­able, but that’s only a rea­son not to do math­e­mat­ics that way if you have an­other way that does an­swer all ques­tions. And those same in­com­plete­ness the­o­rems (more or less) tell us that if what our brains can do is com­putable then there is no way we can do math­e­mat­ics that avoids both in­con­sis­tency and in­com­plete­ness.

• “if what our brains can do is com­putable then there is no way we can do math­e­mat­ics that avoids both in­con­sis­tency and in­com­plete­ness.”

This sen­tence illus­trates the for­mal­ist es­sen­tial­ism that I’m crit­i­ciz­ing. If we con­sider math­e­mat­ics as a so­cial ac­tivity, as Brouwer did, then the no­tion of com­plete­ness doesn’t come up in the first place, and it’s use­less to worry about such a thing. This per­spec­tive, in part, in­fluenced Gödel to make his dis­cov­er­ies in the first place.

Much of the point of Hilbert’s pro­gram (and the wider goal of for­mal­ism/​logi­cism) was to prove math­e­mat­ics in en­tirety con­sis­tent by pro­vid­ing a for­mal logic which could be con­sid­ered math­e­mat­ics it­self. Without that, there’s no mean­ingful sense in which math­e­mat­ics is ac­tu­ally founded on a for­mal logic. After all, that would mean that ev­ery­thing out­side of your cho­sen logic wouldn’t be part of math­e­mat­ics, which is ob­vi­ously wrong. After in­com­plete­ness was es­tab­lished, this situ­a­tion was shown to be ter­mi­nal. I think call­ing the whole pro­ject un­ten­able af­ter the pub­li­ca­tion of Gödel’s in­com­plete­ness the­o­rems is a fairly rea­son­able read of his­tory.

• If what the uni­verse does is com­putable then there is no way the whole com­mu­nity of math­e­mat­i­ci­ans can do math­e­mat­ics that avoids both in­con­sis­tency and com­plete­ness.

Now, of course you’re at liberty not to worry about com­plete­ness. Noth­ing wrong with that. But in that case I don’t see that you can fairly say that for­mal­ism is un­ten­able on ac­count of in­com­plete­ness. If it’s OK not to get an­swers to all math­e­mat­i­cal ques­tions then it’s OK for for­mal­ist math­e­mat­ics not to de­liver an­swers to all math­e­mat­i­cal ques­tions. You might con­tem­plate a strong ver­sion of for­mal­ism one of whose tenets is “all math­e­mat­i­cal ques­tions must be sol­u­ble by these means”, but I claim for­mal­ism shouldn’t be com­mit­ted to that.

I take it your last para­graph is sug­gest­ing that in fact for­mal­ism should be com­mit­ted to that. I dis­agree, or more pre­cisely I think for­mal­ism-with­out-that is prima fa­cie a rea­son­able po­si­tion. I don’t think I un­der­stand what you say about not be­ing “part of math­e­mat­ics”, be­cause (1) some­thing can still be “part of math­e­mat­ics” even if the ax­ioms you’re work­ing with leave it open whether it’s true (one can still prove the­o­rems like “if the ax­iom of choice holds, then X” even if work­ing in a sys­tem that doesn’t de­cide AC) and (2) a for­mal­ist can still choose to work with differ­ent for­mal sys­tems on differ­ent oc­ca­sions, and re­gard both as part of math­e­mat­ics.

• If what the uni­verse does is com­putable then there is no way the whole com­mu­nity of math­e­mat­i­ci­an­scan do math­e­mat­ics that avoids both in­con­sis­tency and com­plete­ness.

I don’t think you un­der­stand what I’m get­ting at. It’s not that com­plete­ness shouldn’t be wor­ried about, it’s that it doesn’t make sense if you aren’t already as­sum­ing that math­e­mat­ics is a for­mal logic. If you worry about for­mal logic then you worry about com­plete­ness. If you don’t as­sume that math­e­mat­ics is a for­mal logic, then wor­ry­ing about math­e­mat­ics does not lead one to con­sider com­plete­ness of math­e­mat­ics in the first place. I’m say­ing that it does not make sense to talk about math­e­mat­ics be­ing com­plete or in­com­plete in the first place, since math­e­mat­ics isn’t a for­mal logic. Yes, it’s im­pos­si­ble for the com­mu­nity of math­e­mat­i­ci­ans to cre­ate a for­mal logic (of suffi­cient ex­pres­siv­ness) which avoids in­con­sis­tency and in­com­plete­ness, but since math­e­mat­ics isn’t a for­mal logic, that doesn’t mat­ter.

“a for­mal­ist can still choose to work with differ­ent for­mal sys­tems on differ­ent oc­ca­sions, and re­gard both as part of math­e­mat­ics.”

Im­plicit in that state­ment is the as­sump­tion that math­e­mat­ics is not, at its core, a for­mal logic. Yes, it con­tains for­mal log­ics, but it isn’t one. I think you’re us­ing a very weak (and very mod­ern) defi­ni­tion of “foun­da­tion of math­e­mat­ics”, be­ing some­thing ca­pa­ble of do­ing a sig­nifi­cant chunk, but not all, of, math­e­mat­ics. I think I’ve been clear in what I mean by “foun­da­tion of math­e­mat­ics”, be­ing some­thing that should be ca­pa­ble of fa­cil­i­tat­ing ALL of math­e­mat­ics. My point is that such a thing doesn’t ex­ist. If you dis­agree, feel free to ar­gue against what I’m ac­tu­ally say­ing.

I do not take is­sue with the idea that one can do a sig­nifi­cant chunk (per­haps most of in prac­tice) math­e­mat­ics us­ing a for­mal logic. That logic would not then be math­e­mat­ics, though. That’s all I as­serted.

I’m get­ting the feel­ing that you didn’t read my post be­cause you’re as­cribing be­liefs to me that I do not hold. I will quote my­self;

[...] many peo­ple seem to think that math­e­mat­ics is, at some level, a for­mal logic, or at least that the ac­tivity of math­e­mat­ics has to be founded on some for­mal logic[...]”

That is the state­ment you’re tak­ing is­sue with, yes? Do you think that math­e­mat­ics is, in fact, a for­mal logic? If not, then you agree with me. Do you think that math­e­mat­ics has to be founded on a for­mal logic? If not, then you agree with me. What are you ac­tu­ally dis­agree­ing with? Are you go­ing to sup­port the as­ser­tion that math­e­mat­ics is a for­mal logic? Are you go­ing to sup­port the as­ser­tion that math­e­mat­ics has to be founded on a for­mal logic?

You might con­tem­plate a strong ver­sion of for­mal­ism one of whose tenets is “all math­e­mat­i­cal ques­tions must be sol­u­ble by these means”

I don’t know what ver­sion of for­mal­ism you think I’m refer­ring to, but my ex­plicit refer­ence to Hilbert should have clued you into the fact that I’m talk­ing about Hilber­tian for­mal­ism. I’d per­son­ally pre­fer it if you didn’t waste time ar­gu­ing with a straw-man.

• You feel like I’m straw­man­ning you. I feel like you’re straw­man­ning me. I pro­pose that we make the ob­vi­ous as­sump­tion that nei­ther of us is de­liber­ately con­struct­ing straw­men (I promise I’m not, though of course you don’t have to be­lieve me) and see if we can come to a bet­ter un­der­stand­ing.

What fol­lows is rather long-winded; I apol­o­gize for not hav­ing had time to make it shorter. I hope I’ve at least been able to make it clear.

1 What is “for­mal­ism”?

“For­mal­ism” can mean a bunch of things. Let me list a few.

F0: Hilbert’s origi­nal pro­gramme of find­ing a sin­gle perfectly for­mal­ized sys­tem, sim­ple enough that no math­e­mat­i­cian could rea­son­ably ob­ject to it, pow­er­ful enough to de­ter­mine the an­swers to all math­e­mat­i­cal ques­tions.

It is (I think) un­con­tro­ver­sial that F0 turned out to be im­pos­si­ble. No one who is ac­tu­ally think­ing about these things is a F0-for­mal­ist now.

F1: The idea that we should pick some sin­gle for­mal sys­tem (maybe ZFC, per­haps aug­mented by some large car­di­nal ax­ioms or some­thing) and say that math­e­mat­ics is the study of this sys­tem and its con­se­quences. (This im­plies, e.g., ac­cept­ing that some math­e­mat­i­cal ques­tions sim­ply have no an­swers. It doesn’t mean that we can’t talk about those ques­tions at all, though; we can still say things like “X is true if Y is” where X and Y are both un­de­cid­able within the cho­sen sys­tem.)

Ob­vi­ously F1 is hope­less if it’s taken to mean that math­e­mat­ics always was pre­cisely the study of the prop­er­ties of ZFC or what­ever, since there was math­e­mat­ics be­fore there was ZFC. But if it’s taken as a pro­posal for how we should cur­rently un­der­stand the prac­tice of math­e­mat­ics, it’s defen­si­ble, and I think quite a lot of math­e­mat­i­ci­ans think in roughly those terms.

(That’s com­pat­i­ble with say­ing, e.g., that later on we might de­cide to switch to a differ­ent for­mal foun­da­tion. And the best way to think about how that de­ci­sion is made might well be in terms of math­e­mat­ics-as-so­cial-ac­tivity. But ad­vo­cates of F1 might pre­fer to say that math­e­mat­ics is a for­mal ac­tivity, but that philos­o­phy of math­e­mat­ics is a so­cial one, and that what hap­pens is that some­times we switch for partly-so­cial rea­sons from do­ing one sort of math­e­mat­ics to an­other sort of math­e­mat­ics.)

F2: The idea that math­e­mat­ics is the study of for­mal sys­tems, of which (e.g.) ZFC is just one. Differ­ent math­e­mat­i­ci­ans might work with differ­ent and mu­tu­ally in­com­pat­i­ble for­mal sys­tems, and that’s fine. Most math­e­mat­i­ci­ans work at a higher level than the un­der­ly­ing for­mal sys­tem, but what makes the stuff they do math­e­mat­ics is the fact that it can be im­ple­mented on top of one of these for­mal sys­tems. The an­cient Greeks didn’t have a de­cent un­der­ly­ing for­mal sys­tem, but what they did could still be lay­ered on top of (say) ZFC and is there­fore math­e­mat­ics.

Ver­sions of F1 that coun­te­nance the pos­si­bil­ity of chang­ing for­mal sys­tem are clearly shad­ing into F2; what dis­t­in­guishes F2 is that F2-ists are com­fortable with the idea that mul­ti­ple differ­ent sys­tems of this type can be around con­cur­rently and equally le­gi­t­i­mate. (Think of it as shift­ing a quan­tifier. F1 says “there ex­ists a for­mal sys­tem P such that do­ing math­e­mat­ics = work­ing in P” and F2 says “do­ing math­e­mat­ics = hav­ing there ex­ist a for­mal sys­tem P such that you’re work­ing in P”. Kinda.)

I don’t claim that F0, F1, and F2 ex­haust the range of things one could call “for­mal­ism”, but they seem rea­son­ably rep­re­sen­ta­tive. And I do claim that all of them can rea­son­ably be called “for­mal­ism”. They both as­sume that what dis­t­in­guishes math­e­mat­ics from other ac­tivi­ties is that it is grounded in for­mal-sys­tem calcu­la­tions. They do, how­ever, both ad­mit that the choice of for­mal sys­tems, and the ex­pec­ta­tion that their calcu­la­tions are worth do­ing, may in turn rest on some­thing else.

I think you may dis­agree with call­ing them “for­mal­ism”, since when I ges­tured ear­lier to­wards some­thing like F1 or F2 you said: “Im­plicit in that state­ment is the as­sump­tion that math­e­mat­ics is not, at its core, a for­mal logic” (etc.). Well, ob­vi­ously what mat­ters isn’t ex­actly what defi­ni­tion we should give to the word “for­mal­ism” but what sorts of po­si­tions math­e­mat­i­ci­ans ac­tu­ally (ex­plic­itly or im­plic­itly) hold, and how co­her­ent and fruit­ful those po­si­tions are.

Many of the dis­cussed philo­soph­i­cal prob­lems, as far as I can tell, stem from the as­sump­tion of for­mal­ism. That is, many peo­ple seem to think that math­e­mat­ics is, at some level, a for­mal logic, or at least that the ac­tivity of math­e­mat­ics has to be founded on some for­mal logic, es­pe­cially a clas­si­cal one.

I don’t think “many peo­ple seem to think” F0, even though that was what Hilbert origi­nally had in mind. And I don’t think F0 is re­quired in or­der for philo­soph­i­cal ques­tions about (e.g.) what num­bers re­ally are to arise. So that’s why I didn’t take you to be talk­ing about F0, but about some­thing more like F1 or F2. In the con­text of what “many peo­ple seem to think”, it seems to me that F0 is it­self a straw­man; not be­cause no one ever em­braced F0 (Hilbert did, and he was no fool) but be­cause so far as I know no one ex­plic­itly does now, and I don’t think any­one does im­plic­itly (in the sense that they think things that only make sense if one as­sumes F0) ei­ther.

It’s true that F1 and F2 al­low for the pos­si­bil­ity that one thing math­e­mat­i­ci­ans do may be to change what for­mal sys­tem they study, and that they will do that on the ba­sis of some­thing not ob­vi­ously re­ducible to for­mal-sys­tem calcu­la­tions. But I can’t agree that that means that they aren’t truly for­mal­ist po­si­tions on ac­count of not mak­ing “ALL of math­e­mat­ics” be about for­mal sys­tems; I think an F1-ist or F2-ist can perfectly well say that while choos­ing a for­mal sys­tem is some­thing a math­e­mat­i­cian may some­times do, mak­ing that choice isn’t math­e­mat­ics but some­thing else closely re­lated to math­e­mat­ics.

2 Is “for­mal­ism” un­ten­able be­cause of the in­com­plete­ness the­o­rems?

F0 is, for sure. F1 and F2, not so much. Some­one may em­brace F1 or F2 but be quite un­trou­bled by the fact that math­e­mat­ics (for an F1-ist) or the par­tic­u­lar va­ri­ety of math­e­mat­ics they hap­pen to be do­ing (for an F2-ist) is un­able to re­solve some of the ques­tions it can raise.

Such a per­son’s po­si­tion would be no­tably differ­ent from yours as I un­der­stand it—you say “it does not make sense to talk about math­e­mat­ics be­ing com­plete or in­com­plete in the first place”; an F1-ist would say it ab­solutely does make sense to talk about that, and as it hap­pens math­e­mat­ics turns out to be in­com­plete; an F2-ist would say that at any rate we can talk about whether a par­tic­u­lar sys­tem we’re work­ing in is com­plete or in­com­plete, and when do­ing math­e­mat­ics we’re always work­ing within some sys­tem and it’s always in­com­plete. (But they might e.g. say that we are always at liberty to use a differ­ent sys­tem, and that if we run across an in­stance of in­com­plete­ness that trou­bles us we can go look­ing for a sys­tem that re­solves it; their po­si­tion might end up re­sem­bling yours in prac­tice.)

3 What do we ac­tu­ally dis­agree about?

Prob­a­bly about whether philo­soph­i­cal as­sump­tions made by “many peo­ple”, to the effect that math­e­mat­ics is fun­da­men­tally about for­mal­ized (or at least for­mal­iz­able) logic, are un­ten­able on ac­count of the in­com­plete­ness the­o­rems. I think it isn’t, be­cause po­si­tions like F1 and F2 in­volve such as­sump­tions and aren’t made un­ten­able by the in­com­plete­ness the­o­rems, and I think those rather than F0 are the po­si­tions held by “many peo­ple”.

Per­haps about whether “many peo­ple” make as­sump­tions that are more or less equiv­a­lent to F0. I think they don’t. Per­haps you think they do.

Per­haps about whether F1 and F2 re­ally say that math­e­mat­ics is fun­da­men­tally about for­mal­ized logic. I think they do. Per­haps you think they don’t.

You asked some ques­tions that (I think) as­sume that I am en­dors­ing “for­mal­ism” in some sense, even if not yours. That’s not what I’m do­ing—noth­ing I’ve said above is in­tended to claim that “for­mal­ism” in any sense is right. (I find F2 tempt­ing, at least, but I’m not sure I would ac­tu­ally en­dorse it.) It is pos­si­ble that it will turn out that I agree with you about, say, whether math­e­mat­ics “has to be founded on a for­mal logic”, while still dis­agree­ing about whether those poor mis­guided souls who think it does are tak­ing a po­si­tion that is un­ten­able be­cause of the in­com­plete­ness the­o­rems.

I don’t know yet whether we dis­agree about the ex­tent to which math­e­mat­ics is a so­cial ac­tivity. Per­haps we do.

• 1 What is “for­mal­ism”?

I think it isn’t, be­cause po­si­tions like F1 and F2 in­volve such as­sump­tions and aren’t made un­ten­able by the in­com­plete­ness the­o­rems

I think it’s ironic that you’re ar­gu­ing with me over the mean­ing of a word, con­sid­er­ing the con­tent of my es­say. I stated at the be­gin­ing of my es­say what I meant by “for­mal­ism”. If you don’t think that word should be used that way, that’s fine, but I’m not in­ter­ested in ar­gu­ing about the mean­ing of a word. By pre­tend­ing that I’m ar­gu­ing against any and all forms of what may be called for­mal­ism, you are re­plac­ing what I ac­tu­ally said with some­thing else. That’s not a sub­stan­tive dis­agree­ment with any po­si­tion I ac­tu­ally en­dorsed.

F1: The idea that we should pick some sin­gle for­mal sys­tem [...] ”

In my origi­nal quote I said “has” for a very spe­cific rea­son. “Should” is a mat­ter of opinion. I don’t think it’s un­rea­son­able to choose a safe win­dow from which to study the uni­verse of math­e­mat­ics, but one shouldn’t speak as if that win­dow is the uni­verse it­self.

I don’t think “many peo­ple seem to think [...]”

When some­one states some­thing of the form “math­e­mat­ics turns out to be in­com­plete” they are as­cribing prop­er­ties of a for­mal logic to math­e­mat­ics. When some­one states that math­e­mat­ics is an ac­tivity in­volv­ing, on oc­ca­sion, a de­ci­sion “to switch to a differ­ent for­mal foun­da­tion”, they are as­cribing prop­er­ties of an ac­tivity which do not hold for for­mal log­ics. This is the cen­tral con­tra­dic­tion I’m fix­at­ing on. When I say “many peo­ple seem to think” I don’t mean that many peo­ple ex­plic­itly en­dorse, but rather that many peo­ple im­plic­itly think of math­e­mat­ics as a for­mal sys­tem. Say­ing “math­e­mat­ics is in­com­plete” is a form of synec­doche, say­ing “math­e­mat­ics” but mean­ing only a part of it. Failure to re­al­ize that this is be­ing done leads peo­ple to say silly things.

F2: The idea that math­e­mat­ics is the study of for­mal sys­tems

mak­ing that choice isn’t math­e­mat­ics

A field of study can’t be in­com­plete in the way a for­mal logic can. Say­ing “math­e­mat­ics is in­com­plete” is in­com­pat­i­ble with the view that math­e­mat­ics is a field of study, and yet I’ve seen many peo­ple en­dorse such a view. If you say that math­e­mat­ics is the study of for­mal sys­tems, I’d say that’s wrong, but that’s not rele­vant to any of my ear­lier points.

I think this might ac­tu­ally be the main point of dis­agree­ment. Mak­ing that choice in­volves math­e­mat­i­cal rea­son­ing and in­tu­ition which is cer­tainly part of math­e­mat­ics, not least be­cause it’s part of what math­e­mat­i­ci­ans, in par­tic­u­lar, ac­tu­ally do. Ex­clud­ing such things from be­ing math­e­mat­ics is ar­bi­trary and ar­tifi­cial. If you’re go­ing to make such a des­ig­na­tion, then it seems the ul­ti­mate goal is to make math­e­mat­ics mean “the study of for­mal sys­tems”, but I have no in­ter­est in talk­ing about such a thing. This is, again, ar­gu­ing over a defi­ni­tion.

In­ci­den­tally, I stated that the po­si­tion which was un­ten­able af­ter Gödel’s In­com­plete­ness The­o­rems is the as­sump­tion, im­plicit in the state­ment “math­e­mat­ics is in­com­plete”, that math­e­mat­ics is a for­mal logic. How­ever, that doesn’t ap­pear as ei­ther your F0, F1, or F2.

I’ve found this dis­cus­sion to largely be a waste of time. I won’t be re­spond­ing be­yond this point.

• I am sorry that you haven’t found the dis­cus­sion use­ful. For my part, I am also dis­ap­pointed by how it’s turned out, and es­pe­cially by how ready you seem to be to as­sume bad faith on my part.

Since ob­vi­ously you don’t want to con­tinue this, I won’t re­spond fur­ther ex­cept to cor­rect a few things that seem to me to be sim­ply er­rors. One: I am not (de­liber­ately, at least) “pre­tend­ing” any­thing, and in par­tic­u­lar I am not “pre­tend­ing that [you’re] ar­gu­ing against any and all forms of what may be called for­mal­ism”. I thought I went out of my way to avoid mak­ing any claim of that sort. What I am claiming is that the things you said about “many peo­ple” ap­ply only to “weaker” ver­sions of for­mal­ism, while at least some of the ob­jec­tions you make ap­ply only to “stronger” ver­sions. The point of list­ing some par­tic­u­lar ver­sions was to try to clar­ify those dis­tinc­tions. Two: Once again, al­though I am dis­cussing and to some ex­tent defend­ing some kinds of for­mal­ism, I am not en­dors­ing them, which much of what you’ve writ­ten seems to as­sume I am. Three: the po­si­tion you ac­tu­ally said was un­ten­able be­cause of the in­com­plete­ness the­o­rems was “that math­e­mat­ics is, at some level, a for­mal logic, or at least that the ac­tivity of math­e­mat­ics has to be founded on some for­mal logic, es­pe­cially a clas­si­cal one” (em­pha­sis mine), and it still seems to me that all of F0, F1, F2 say pretty much that.

Ac­tu­ally, I will say one other thing, though I’m not ter­ribly op­ti­mistic that it will help. The main point I’ve been try­ing to make, though per­haps I haven’t been as ex­plicit about it as I should, is that I think you are as­cribing to “many peo­ple” a po­si­tion more ex­treme, and sillier, than they would ac­tu­ally en­dorse, and that the bits of that po­si­tion that lead to bad con­se­quences are ex­actly the bits they wouldn’t ac­tu­ally en­dorse. E.g., the idea that math­e­mat­i­ci­ans do noth­ing other than for­mal ma­nipu­la­tion (of course they don’t, and ev­ery­one knows that, and no I don’t think the things peo­ple say about for­mal sys­tems im­ply oth­er­wise). Or the idea that if some­one says “math­e­mat­ics is in­com­plete” this means that they don’t know the differ­ence be­tween a field of study and a for­mal logic, rather than that they are say­ing that we should think of the prac­tice of that field as in prin­ci­ple re­ducible to op­er­a­tions in a for­mal log­i­cal sys­tem, which is in­com­plete. Etc.

• >If you don’t as­sume that math­e­mat­ics is a for­mal logic, then wor­ry­ing about math­e­mat­ics does not lead one to con­sider com­plete­ness of math­e­mat­ics in the first place.

To make sure I un­der­stand this right: This is be­cause there are definitely com­pu­ta­tion­ally-in­tractable prob­lems (e.g. 3^^^^^3-digit mul­ti­pli­ca­tion), so math­e­mat­ics-as-a-so­cial-ac­tivity is ob­vi­ously in­com­plete?

• No. I’m not ad­vo­cat­ing for some sort of fini­tism, nor was Brouwer. In fact, I didn’t ac­tu­ally men­tion com­putabil­ity, that’s just some­thing gjm brought up. It’s ir­rele­vant to my point. Math­e­mat­ics is a so­cial ac­tivity in the same way poli­tics is a so­cial ac­tivity. As in, it’s an ac­tivity which is so­cial, or at least pred­i­cated on some sort of so­ciety. Say­ing that math­e­mat­ics is in­com­plete is as mean­ingful as say­ing that poli­tics is in­com­plete in the same way a for­mal logic might be. It just doesn’t make sense.

Note that the in­tu­itions which jus­tify the us­age of a par­tic­u­lar ax­iom is not part of an ax­iom sys­tem, but those in­tu­itions would still be part of math­e­mat­ics. That’s largely the in­tu­ition­ist cri­tique of “old” for­mal­ism. It was also used as a cri­tique of log­i­cal pos­i­tivism by Gödel.

• I get that old for­mal­ism isn’t vi­able, but I don’t see how that ob­vi­ates the com­plete­ness ques­tion. “Is it pos­si­ble that (e.g.) Gold­bach’s Con­jec­ture has no coun­terex­am­ples but can­not be proven us­ing any in­tu­itively satis­fy­ing set of ax­ioms?” seems like an in­ter­est­ing* ques­tion, and seems to be about the com­plete­ness of math­e­mat­ics-the-so­cial-ac­tivity. I can’t cash this out in the poli­tics metaphor be­cause there’s no real poli­ti­cal equiv­a­lent to the­o­rem prov­ing.

*In­ter­est­ing if you don’t con­sider it re­solved by Godel, any­way.

• Math­e­mat­ics is a so­cial ac­tivity in the same way poli­tics is a so­cial ac­tivity. As in, it’s an ac­tivity which is so­cial, or at least pred­i­cated on some sort of so­ciety.

Are you sayig that noth­ing a her­mit would ever do can be called math­e­mat­ics? That doesn’t seem right.

• I’m pretty sure An­thony isn’t claiming that math­e­mat­ics is so­cial in the sense that ev­ery math­e­mat­i­cal ac­tivity in­volves mul­ti­ple peo­ple work­ing ac­tively to­gether. But that her­mit would be do­ing math­e­mat­ics in the con­text of the math­e­mat­i­cal work other peo­ple have done. Sup­pose the her­mit works on, say, the Rie­mann hy­poth­e­sis: they’ll be build­ing on a ton of work done by ear­lier math­e­mat­i­ci­ans; the fact that they find RH im­por­tant is prob­a­bly strongly in­fluenced by ear­lier math­e­mat­i­ci­ans’ choices of re­search top­ics; the fact that they find other things RH re­lates to im­por­tant, too. Sup­pose they think of what they’re do­ing in the con­text of, say, ZFC set the­ory; there are lots of pos­si­ble set-the­o­retic foun­da­tions (note: An­thony would prob­a­bly pre­fer to avoid this no­tion of “foun­da­tions”) one could use, and the par­tic­u­lar choice of ZFC is surely strongly in­fluenced by the foun­da­tional choices other math­e­mat­i­ci­ans have made.

(An­thony might per­haps also want to say, though here I don’t think I could agree, that if e.g. the her­mit writes proofs then what those look like will be largely de­ter­mined by what sorts of ar­gu­ments math­e­mat­i­ci­ans find con­vinc­ing: that the point of a proof is pre­cisely to con­vey ideas and their cor­rect­ness to other math­e­mat­i­ci­ans, which is a so­cial ac­tivity even if those other math­e­mat­i­ci­ans hap­pen not to be there at the time. I don’t agree with this be­cause I think our her­mit might well pur­sue proofs sim­ply for the sake of en­sur­ing the cor­rect­ness of their con­clu­sions, and a suffi­ciently smart her­mit might come up with some­thing like the no­tion of proof all on their own with only that mo­ti­va­tion.)

• That’s a pretty low bar. Is wiping your ass a so­cial ac­tivity too? Be­cause, pre­sum­ably, your mom taught you how to do it, and the fact you’re do­ing it with pa­per is strongly in­fluenced by ear­lier ass wiper’s choices.

But never mind that. Sup­pose the her­mit never learned any math, not even ad­di­tion. Will you say that his math would still be so­cial, be­cause he already knew the words “zero”, “one”, “two”, which hint at the set of nat­u­rals? Then sup­pose that the her­mit has not seen a hu­man since the day he was born, was raised by wolves, de­vel­oped his own lan­guage from zero, and then de­scribed some the­ory in that (in­deed, this her­mit might be the great­est ge­nius who ever lived). Surely that’s not so­cial. But is it not math?

• Per­son­ally, I’d be perfectly happy to say that our hy­po­thet­i­cal her­mit is do­ing math­e­mat­ics de­spite the com­plete ab­sence of so­cial con­nec­tions; but I wasn’t en­dors­ing the claim that math­e­mat­ics is a so­cial ac­tivity, merely ex­pli­cat­ing it. (And of course it’s pos­si­ble that my ex­pli­ca­tion fails to match what An­thony would have said.) I am not con­fi­dent enough of my un­der­stand­ing of An­thony’s po­si­tion to guess at his an­swer to your hy­po­thet­i­cal ques­tion.

(But, for what it’s worth, if for some rea­son I were re­quired to defend the math­e­mat­ics-is-so­cial claim against this ar­gu­ment, I think I would say that it suffices that math­e­mat­ics as ac­tu­ally prac­ticed is so­cial; mak­ing poli­ti­cal speeches is fairly un­con­tro­ver­sially a so­cial ac­tivity even though one can imag­ine a su­per­ge­nius her­mit con­tem­plat­ing the pos­si­bil­ity of a so­ciety that fea­tures poli­ti­cal speeches and mak­ing some for fun.)

• Un­like grounded in­tu­itions, an un­grounded one may be such that it’s never mod­ified by new in­for­ma­tion. This doesn’t de­scribe all un­grounded in­tu­itions, but it de­scribes the ones we’re in­ter­ested in.

I think my first in­tu­ition of “set” was mod­ified by ob­serv­ing Rus­sell’s para­dox.