So my fiance Sharon Berry posted a very early (like 2 years early) draft of her thesis on a wiki here. The broad question she is addressing is how we can come to have accurate mathematical knowledge which I figured might be of interest to some people who check out the philosophy part of my blog. I also figured I’d take this chance to share my own thoughts on the subject. However, to give credit where credit is due I would never have really thought through these issues if Sharon hadn’t brought up the subject and many of the ideas are really hers. However, I take them in a very different direction than she does.

The really short version of my attitude to the problem of mathematical knowledge is “What Problem?” I mean obviously mathematical knowledge is subject to the same skeptical doubts that other forms of knowledge are but I’m unconvinced that there is any particular problem unique to mathematical knowledge. More specifically I would say that mathematical knowledge is nothing but a limiting case of other sorts of knowledge so it poses no problem over and above the problem of understanding the meaning and our knowledge of other sorts of statements. Of course explaining meaning is a notoriously difficult problem in it’s own right but I’m tempted to think that it’s a hopeless problem. Ultimately one must merely take meaning to be a primitive concept but that’s another discussion.

I need to get back to working on my thesis so I won’t give more than a very very quick sketch of my thoughts here but roughly I take it there are two primary reasons one might think that mathematical knowledge requires special explanation.

1. The Benacerraf problem: How could we come to know anything about numbers if they don’t have causal powers, we don’t interact with them and so forth.
2. How could it be that our mathematical theories turn out to be useful in the way they are.

Platonism and Reference

So if one accepts a platonic theory of mathematical meaning then there may indeed be special problems about mathematical knowledge. That is if the meaning of a statement like 2+2 =4 is really that some special 2 object out there bears a certain relation to itself and the four object one might wonder how it is that we come to know about these platonic objects. However, I’m inclined to simply turn the question around and ask whether the platonic theory in question provides any reason to think that “2″ refers to something we would ‘recognize’ as an integer or whether it could (logically not metaphysically) be that 2 refers to the concept of bunny rabbits and all our statements about arithmetic are really nonsensical. If the platonic interpretation of mathematics tells us that the reference of two must really behave like 2 to qualify as the correct reference then we know exactly how we come to have true beliefs about the numbers — because if our beliefs weren’t largely true we would be talking about something else[^enough]. On the other hand if we don’t have any restrictions about what sort of platonic object 2 might refer to then we aren’t justified in adopting this kind of theory in the first place.

Unfortunately the debate about Platonism and competing philosophies of mathematical largely distracts from what I think are the important issues. As I’ve argued previously Platonism in and of itself says very little about mathematics. What the last paragraph as well as my previous post on the issue emphasize is that it isn’t really Platonism that is doing the work it is your theory of reference. Really on it’s own Platonism says nothing very significant¹, it’s the means by which our talk maps to particular platonic objects that really does the work in the theory. This raises the obvious question of what we even mean when we say that the reference of a certain term is such and such. Are we merely making a claim about dispositions and talk or are we invoking some real metaphysical relation. While Platonism provides a good motivation to consider the issue I think a proper examination of this question of what sort of thing the meaning relation is in the first place illustrates the non-problem of philosophy of mathematics in general.

Platonic Realism About Reference

There are two ways one could understand claims about meaning and reference One could think that the relation of meaning is a truly objective notion with metaphysical substance. That is that the relation between words/mental states/speaking contexts is some and references/meanings is something like a platonic entity in it’s own right. On such a theory it is presumably logically possible (but not metaphysically) that when I say “2″ it really (by virtue of this objectively existing meaning relation) refers to rabbit. In other words the meaning of word is a notion much like the moral status of an action under on a realist moral theory.

Just as with moral realism I think the appropriate response to this notion of meaning is to challenge that it counts as meaning at all. Ultimately there is just this relation out there mapping situations/worlds/utterances/mental states/whatever to references/intentions but why should we think this picks out what we talk about when we use the term meaning? Additionally on this sort of platonic realism about meaning we don’t have any reason to actually believe that we really do have knowledge. After all maybe the objective meaning relation isn’t what we think it is at all and what we take to be true statements aren’t true at all.

One might still be tempted to insist that obviously we have knowledge thus the fact that this theory can’t explain this fact is a puzzle requiring explanation. However, this simply gets things backwards and implicitly rejects the very assumptions of the theory itself. If we accept this sort of theory we need to just bite the bullet and say we don’t know if we really know anything and thus how we know things doesn’t require explanation. Personally I think our intuition that our usage determines meaning is a good reason to reject this sort of theory but in either case this leaves no special problems for mathematics. Of course you might try and say that the mapping between statements and meanings/references must obey certain restrctions but this does no good at all since of course any actual map will have some facts that are true of it but this does nothing to offer us reason to think we have any knowledge of what they are.

Naturalist Theories of Meaning

I think a much more promising approach² is to jettison all the metaphysical baggage and start from the assumption that meanings, ultimately must be defined in terms of sounds, dispositions, actions and other arrangements of matter. That is nothing special or magical goes on with meanings. They are just a concept introduced to organize very complicated descriptions of human behavior in terms of atoms and physical laws. Thus the ultimate standard against which we judge a theory of meaning is it’s predictive accuracy and theoretical utility (how well does it work with other models we wish to use). In some sense already this approach should suggest that there shouldn’t be any deep paradoxes in terms of meaning. After all we are confident that the description of human behavior at the level of atoms is consistant thus any apparent difficulty at the level of meanings either reflects a confusion on our part or a poor choice of definition.

To put the point a bit differently we should think about a theory of meaning much the way we think about thermodynamics as derived from statistical mechanics. Yes, it can be a powerful theory with useful concepts and important impacts but ultimately just as debates about whether entropy is the log of the number of possible states holding X,Y and Z fixed or just X and Y doesn’t reflect any fundamental fact about the universe but a definitional choice we make that is judged on it’s utility. Thus theories like fictionalism or formalism shouldn’t be understood as making different philosophical claims but rather judged simply on their utility in predicting how people actually use words. Indeed one might very well conclude that different models are most appropriate in different circumstances.

Ultimately then the question about how we can come to have mathematical knowledge is largely a non-question. I can point to the actual ways that mathematicians prove theorems and reach conclusions and that right there shows how we come to have mathematical knowledge. Still one might ask but why are the results of our proofs actually true? However, this has a totally trivial answer. The reason that proofs give us true mathematical results is that every step of the proof is truth preserving. Indeed we can go through this and using the fact (in the meta-language) that A and B is true if and only if A is true or B is true show that the methods mathematicans use to reach theorems really do produce true theorems. Asking for anything more is a demand to know why logic is true. Obviously at a very basic level we have to just assume that logic is true (see Quine’s arguments about this point in his discussions of radial translation) so it’s unclear what is left to be explained at all.

To put the point slightly differently it’s contradictory to worry about how we get mathematics correct. Either the question tells us how we have reason to believe we do get mathematics right, in which case it tells us the answer or it offers no such explanation and we have no need to explain a phenomena that we don’t have reason to accept as true.

Usefullness of Mathematics

This finally brings us to the question of why mathematics turns out to be useful. One might think that it’s surprising that the results of mathematics tells us useful things about the world. Certainly in one sense it is surprising, but that’s the sense in which the understandability of the world is surprsing, i.e., that induction works. While it may appear that mathematics directly makes predictions about the world (if I have two apples in my bag and place another two apples into my bag I have four apples in my bag) in fact it’s only the combination of mathematical theorems with contingent bridge laws that makes these predictions (apples don’t appear or disappear when I place more of them together). One might try and minimize the significance of these bridge laws by saying something like “So long as apples don’t appear or disappear the number of apples in my bag is the number of apples I added minus the number I removed.” However, this merely begs the question by working in our expectation that the plus operation on the natural numbers describes how objects behave into the definition of appear or disappear. I could equally well claim that apples were disappearing and reappearing all the time but if they didn’t do so we could see that adding n apples to a bag with m apples in it results in a bag with n x m apples in it.

In fact the usefulness of something like mathematics is an easy consequence of a well known theorem in recursion theory. Supposing we have a language complex enough to express arbitrary procedures³ then that language will contain infinitely many different ways to state the same procedure, some subset of which will be possible to construct a verification that they are equivalent. In other words no matter how weird your language is you can’t get around the fact that some things will turn out to be non-obviously equivalent which suggests that it will be useful to have a means to identify at least some of them.

Usefullness and Knowledge

The final worry is that one might try and link the two concepts and ask how it is that we come to have mathematical knowledge that yields useful results. Thus even if we don’t have abstract reasons to believe that the syntactic manipulations of mathematicians meet some independent standard of being true we do notice that they let us build rockets and cure disease and the like. Thus one might think the utility of mathematics requires some explanation.

Once again though I think a careful examination of the question reveals it to be a non-worry. If by mathematics you merely mean the sort of thing that mathematicians do then it’s undeniable that what counts as mathematics is partially determined by what is useful. While many types of mathematics are very abstract the subject in the large is influenced by what has solved problems presented by the world. This point is made even more forcefully if you try to define mathematics as any abstract rule based manipulation of symbols. After all under such a definition certain types of astrology would qualify which most assuredly is not useful. Similarly any other means by which you tried to formally define the problem is likely to either reduce to triviality or not call our for any explanation at all.

This was a pretty hurried and scattered explanation of my thouhts so hopefully people ould follow it. If you are confused but curious about what I’m trying to say anyway feel free to post a comment or ask me via email

UPDATE: Fixed some wording, added clarification at the bottom.

Consider a philosophical debate between two materialists over whether a virus is alive. Philosopher A advances the hypothesis that any organism capable of manipulating it’s environment to copy itself is alive. Philosopher B counters that Shakespeare’s “Macbeth” has this property as it’s induces people to produce reproductions of the work and instead argues that a being is alive only if it doesn’t require outside intervention to duplicate itself. Philosopher A counters with an example of a plant that humans have cultivated for long enough that it is now incapable of reproduction without intentional human intervention.

Now we can imagine this debate continuing as both philosophers refine their theories as to what constitutes life but no matter how long they argue in this fashion they can’t hope to reach any substantive conclusions. Why? Because they didn’t disagree about anything but word usage at the outset. As materialists they both reject the notion of some elan vital that we might add to our fundamental ontology to ‘explain’ what counts as alive and what doesn’t. As far as the virus goes they would both accept the biologists explanation as to how it reproduced they only disagree on how to label this event. Now there are certainly times it makes sense to argue about labeling but we naturally expect such debates to take a very different form. In particular when people merely disagree about how we should term something they will usually sidestep the debate eventually and simply qualify their wording. When people disagree on how in fact people are inclined to use words they tend to either shrug and move on or to start pulling out real empirical data¹. In no case where people understand themselves to be merely debating a matter of labeling would we expect them to argue about the issue for decades in reputable journals with no hint that they view themselves as debating some empirical fact about usage or pragmatic fact about what would make for good usage.

Unfortunately there seems to be no shortage of arguments in philosophy that can’t be explained as anything other than a confusion of a question of terminology as a substantive question². There are a host of examples but let me give a couple

Is formalism or fictionalism the right philosophy of math?

Now there is (arguably) a genuine substantive question as to whether mathematical Platonism is true. Is there or is there not a realm of platonic objects out there? And if you believe in a substantive notion of reference (reference facts aren’t ontologically reducible to physical/mental facts) whether or not that is what we refer to with mathematical talk. However, when we get down to debates between functionalism and fictionalism things suddenly become much more unclear.

Neither theory disagrees about what in fact mathematicians assert nor makes any fundamentally different metaphysical suppositions. Nor do the two theories compete on genuine empirical predictions. Neither theory attempts to best predict what in practice people will tend to assert about mathematics. So in what sense can there be said to be a substantive question at issue here? And if not why believe these two interpretations are in opposition to each other?.

The “Proper” Conception of Evidential Support

In formal epistemology there seems to be a great deal of ink spilt arguing over what the ‘proper’ notion of confirmation is. Now Brandon Fitelson has always (at least to my eyes) pushed for a ‘there is no fact of the matter’ resolution to many debates in this area so I wouldn’t accuse him of making this sort of mistake but this paper of his gives a nice picture of what sorts of arguments are at play in the area. In particular there seems to be a long lasting dispute as to what the ‘right’ notion of confirmation turns out to be. Is it a three place relation between evidence, theory 1 and theory 2 or is it a two place relationship between evidence and a hypothesis?

Now I would be most surprised if anyone in this debate thought confirmation was an substantive notion (but I’ve been wrong about this sort of thing before), that is that when we assert that evidence E favors hypothesis H we aren’t just asserting some fact about probabilities, models or events but actually claiming that there is some special ‘confirmation’ property in our ontology that adheres to that particular relation but not to others that we might have chosen instead to term confirmation and that. Yet if we aren’t being ontologically liberal like this it would seem that all this debate about what is the ‘right’ notion of confirmation seems silly. We can all agree on the formal consequences of each notion and just set aside the contentious terminological question³.

Is Welfare Desire Satsifaction or Utility Maximization

I mention this because it was the argument that first made me realize that many of these debates couldn’t be substantive. While many people want to add an extra ontological fact to explain morality few people are inclined to indiscriminately add ontological entities for subsidiary moral concepts like welfare yet they are perfectly happy to debate the issue as if it were substantive.

In particular many people argue about whether we maximize welfare by maximizing utility or by maximizing desire satisfaction (or something else) as if it was a separate question we resolve prior to figuring out what is morally good. However, short of proposing a new fundamental property or relation it would seem that the debate over what increases someone’s welfare is merely a terminological question.

Kripke’s Causal Theory of Reference for naturalists

I debated about including this one since some people who buy into Kripke’s theory believe it as a genuine substantial claim. That is they add extra fundamental objects to their ontology (references, meanings etc..) and make the substantial claim that somehow our intuitive explanations of words in terms of other words track these objects and that as a real ontological fact it turns out that the reference of our word is determined by it’s causal history.

However, many of the people who take these theories seriously would call themselves naturalists or physicalists and would balk at the suggestion that by endorsing Kripke’s theory they were making grandiose metaphysical claims that couldn’t possibly be explained in terms of anything physics could in principle ever discover. Presumably as a physicalist one should accept the fact that there is nothing more to speech acts than certain configurations of matter and that there is no free floating metaphysical object ‘the reference’ that exists over and above the configuration of matter. As I’ve argued before it’s absurd for a good naturalist/physicalist to have a horse in the internalist/externalist debate except insofar as one turns out to be a genuinely better empirical predictor of future events.

My thesis is that there is a strong bias towards taking mere disputes about terminology and assuming that they are substantive. Not only is it a tempting fallacy to fall into on it’s own but it also creates for a much more interesting seeming discussion. It seems much more significant to say that one is figuring out the nature of life than to admit one is merely debating what we should call “life.” In any case whatever the reason it seems that this is a common fallacy that I see in philosophical discourse and one we should guard against. There are more than enough substantive arguments to keep philosophers busy and some of these non-substantive arguments are worth having as well but which arguments we take to be persuasive will be very different once we understand it is merely a terminological debate. Importantly once we accept that many of our debates are merely terminological we can no longer assume that there is any tension between things like internalism and externalism or different measures of confirmation.

As an aside I think this is in some sense the issue between Carnap and Quine over the nature of analyticity but that’s something for another post.

CLARIFICATION: I don’t want to claim that these debates couldn’t be rendered substantive. Really all I want to claim is that they are not naively substantive questions so using a standard of argumentation suited to this assumption is in error. I don’t mean to say that we need to abandon these questions only that we should figure out what we are trying to say and what it would take to establish our claim before we try to argue for one side or another.

Also I’m not convinced that any particular philosopher is making this error. It often seems that when I talk to any given philosopher about the matter they have some complex alternative interpretation of the claims at issue that either recognizes them as not substantive or renders them so through some non-obvious interpretation. It’s entirely possible that this is merely a process error but at the very least something is wrong when people adopt the form of a substantive argument for notions that don’t seem like they could be substantive without giving an indication as to what way it is (non-obviously) substantive (least different people think they are debating different questions). What I really want to do hear is not so much to advance my particular theory as to what is going wrong but to call attention to the fact that something seems really out of wack (or have someone give me a satisfactory explanation as to why it is not).

In an earlier post post I reviewed the mathematical significance of Tarski’s definition of truth and promised that I would explain why it is philosophically useless in a later post. I wasn’t sure anyone was interested and never got around to making that post but thanks to a comment I’ve finally gotten around to this post.

Briefly Tarski’s ‘defines’ truth by endorsing the following scheme (called the T-schema) for every sentence S:

‘S is true’ if and only if S

To take the canonical example:

“‘Snow is white’ is true” if and only if ‘Snow is white’

Now I don’t dispute that this scheme indeed holds (perhaps even necessarily) for the concept truth. I’m even willing to grant that this scheme may ‘define’ truth in some sense but what I dispute is that it tells us anything significant about truth.

Although we normally expect a definition to explain¹ the concept in question not all definitions do so. For instance consider the definition of a ‘good action’ as “an action one morally ought to take.” While such a definition might explain the term ‘good action’ to someone who didn’t speak the language or help organize a philosophical theory in no way could it be said to explain what a good action was or give us a grip on the nature of morality. That is the definition merely reveals trivial linguistic connections between words that all make use of the same underlying opaque concept.

Tarski’s definition of truth is just the same. Sure if one didn’t know how to translate the English word ‘true’ into German communicating the T-schema in English could help your listener realize that ‘wahr’ was the correct translation. However, the T-schema only manages this by bootstrapping off the fact that your listener already understands assertability in English. In other words all the T-schema expresses is the connection between the predicate ‘true’ and assertability. In order to find the fact that “‘Snow is white’ is true” if and only if ‘Snow is white’ enlightening you must already understand what it means for snow to be white. If you don’t see this just imagine a linguistic practice where uttering ‘Snow is white’ conveyed your disbelief in the whiteness of snow. In that case the the word defined by the T-schema would have the same meaning as our word ‘false’.

When philosophers ask “What is truth?” they don’t want to know when they should term a sentence ‘true’ supposing they already know when to endorse it as fact. Rather they want to know what it means to assert something as fact, e.g., what is the relationship between ‘Snow is white’ and the external world. Tarski’s ‘definition’ of truth does nothing to explain truth in this sense, it is up to other theories like the correspondence theory of truth to answer this question. The T-schema (in a philosophical context) only illustrates the obvious and trivial relationship between asserting a sentence and asserting that sentence is true. As I outlined in my previous post on the subject Tarski’s definition of truth is a substantial mathematical contribution but it just doesn’t cut much philosophical ice.

Note that nothing I’ve said here conflicts with the deflationist position that the T-schema is the only thing that can be cogently said about truth. I actually find this position quite appealing but the substance of this view is not that the T-schema says something substantial about truth but that there isn’t anything substantial to say about truth. Intuitively this seems right because you must already understand what it means to assert things to even make sense of any other proposal but I haven’t thought about it enough to be totally convinced.

So today in my philosophy seminar we talked about Tarski’s ‘definition of truth.’ This topic never fails to get my blood boiling because philosophers just can’t seem to avoid ascribing great philosophical importance to what is really a totally trivial (philosophically) statement. I don’t know if it is reflexive respect for hard subjects like mathematics, simple confusion caused by the subtle distinctions involved, or just the misleading terminology but, like some kind of philosophical crack, no matter how many times philosophical uses of Tarski’s ‘definition of truth’ are slapped down someone always comes back for another hit. So in case you don’t already understand why Tarski’s definition of truth can do no serious philosophical work I plan to rant about it here, but not in this post. Since many philosophers seem to get blinded by the math in this post I will try and explain the mathematical background. Hopefully this will be generally understandable but if you haven’t seen the T-schema before I suggest you check out the wikipedia article on the semantic theory of truth and the Stanford Encyclopedia of Philosophies entry on Tarski’s Truth Definitions. The more intrepid reader may also want to check out a brief outline of Tarski’s paper or for those of you with JSTOR access Tarski’s 1944 paper.

In the following paragraphs I will be employing the notions of a model. Unfortunately this is a necessary technical notion but a definition can be found over on wikipedia. In case you don’t want to parse the formal definition the basic idea is that a model is a mathematical world or state of affairs. One way to think of it is something like an ersatz possible world and just like a possible world things are true or false in a model (don’t take the analogy too far, mathematical truth is necessary after all). A model is a possible way certain relations, functions and constants could work together and statements are to be true in a model just if they describe the ways things really are in that model. Terminologically saying M models a sentence s just means s is true in M, also written M |= s (M models s).

For instance we might have a model M in a language with one unary relation symbol R and five constant symbols a,b,c,d,e. The model in this case just tells us which constants the relation R holds of. So in this simple case we might specify M by saying R_M holds of a,c,d and doesn’t hold of b,e. Then for this example of M we would have M |= “R(a) & R(d)” but M wouldn’t model “R(b)”. Also, if we pretend that A is really upside down (forall symbol), v denotes or and ~ not then M |= “Ax(R(x) v ~R(x))” (trivial tautology) but M doesn’t model “Ax(R(x))”. Note that a model (unlike just a list of sentences) has to give a consistent account of the way things work. One cannot both have M |= “R(a)” and M |= “~ R(a)” (unless we start talking about para-consistent logics or other silly things).

So the basic mathematical problem which Tarski’s definition of truth solved is something like the following (rephrased in terms of set theory instead of the more abstruse higher order logics Tarski initially had in mind). We are working inside set theory (the meta-language) and wish to define a formal language (the object language) and truth for that formal language. In particular say we want to talk about sentences in the language of number theory (L(N) ) and models (or structures if you prefer) in that language. Now defining truth for sentences in the language of number theory means giving a definition for the relation M |=_L(N) s (M models s) where M is a model in the language of number theory and s is a sentence in that language. Such a definition is necessary because formally a sentence in the language of number theory is just a certain sequence of symbols satisfying syntactic constraints and only by defining a notion of satisfaction can we give it semantic meaning. Of course such a definition should reflect our intuitive idea of what it means to be true, i.e., if 2_M +_M 2_M = 4_M then we should have M |=_L(N) “2+2=4″.

Now if all we want to do is define the set of sentences which is true in some particular model M there is no problem. M is a set and M gives us the atomic sentences which are true in M so by normal set theoretic methods we can get a set T(M) (the set of true sentences in M). Where things get complicated is if we want to define a relation |=_L(N) which holds for any model M and sentence s in L(N). Since the collection of models in L(N) is a proper class there is no set giving us the relation |=_L(N). We are stuck with the fact that |=_L(N) is a proper class and if we want to work conveniently with that proper class we would like to find a formula |=_L(N)(M, s) (slight abuse of notation using the symbol to denote both the formal definition and the relation) in the language of set theory (meta-language) which holds iff M |=_L(N)s. Tarski’s definition of truth lets us build such a formula, or in other world it lets us define truth (at least for first order languages) in a formal language by a finite list of conditions (we can also get rid of the explicit relativization to a particular language and make it a three place relation between a model, sentence and language).

Tarski’s key insight is that we can use a recursive definition to define truth. In particular assuming we have chosen to let the symbol & in our object language (L(N) in this example) play the same semantic role as ‘and’ in our meta-language (language of set theory) then for any well formed formulas x and y in the object language we can define M |=_L(N) “s & t” as follows:

M |=<sub>L(N)</sub> "s & t" iff M |=<sub>L(N)</sub> "s" and M |=<sub>L(N)</sub> "t"

The same trick can be played with all our logical connectives and functional symbols giving us a definition for truth in M. Of course such a definition only works if we somehow know for what atomic (can’t be broken up into smaller components) sentences x, M |=_L(N) “x”. However, in the case of truth in a model these facts can be read directly off of the model. Thus we end up with a finite definition of the relation |=_L(N) which looks something like this:

M |=<sub>L(N)</sub> r iff

    1) r = "s & t" and M |=<sub>L(N)</sub> "s" and M |=<sub>L(N)</sub> "t"
    OR
    2) r = "s v t" and M |=<sub>L(N)</sub> "s" or M |=<sub>L(N)</sub> "t"
    OR
    3) r = "(Ax) s" and ....
    .
    .
    .
    OR
    LAST) r is atomic and M says r is true 
    (for atomic sentences this can be read right off the model) 

Importantly nothing in Tarski’s solution in this case presupposes that M is a set, we could just as well substitute in the defining formula of a proper class.

The philosophical usage of Tarski’s definition of truth is really is sorta special case of the definition of truth in a model I gave above. The special case is where the “model” in question is actual truth, i.e., the same universe we are evaluating our meta-language in. Technically we are no longer really evaluating truth in a model but the basic idea stays the same. We can still define the semantics of our object language via a finite recursive definition in our meta-language except now the mentions of our model are just replaced with actual truth. So for instance the example given above becomes (where “s & t” is a statement in our object language”.

"s & t" is true iff "s" is true and "t" is true

The rest of the definition can be filled out similarly. Though of course this only defines truth if we presuppose we know when atomic sentences in our meta-language are true. Now if we suppose that the object language is actually a fragment of our meta-language and the symbols are to have the same meaning in both, so ‘and’ in the object language is meant to have the same semantic meaning as ‘and’ does in our meta-language, we can follow through the definitions and get Tarski’s famous T-schema.

T: "P" is true iff P

Where “P” is a statement in the object language and P is a statement in the meta-language. For example one instance of this schema is the following.

"Snow is White" iff Snow is White.

However, unlike in the formal case in natural language we no longer have a restricted finite list of logical symbols (symbols like & which are interpreted as part of the logic). Thus the definition of truth is no longer even a finite definition in terms of the atomic truths. Additionally it is unclear what use such a schema could have in natural language. Unlike the formal case it isn’t clear that there is a special role to play for finite syntactical definitions. Most importantly though the utility of Tarski’s definition of truth is that it lets us finitely define truth in an uninterpreted object language given an already understood meta-language. Hopefully you already have suspicions about the philosophical utility of such a ‘definition’ and in my next post I will give a full fledged argument as to why it doesn’t tell us anything we don’t already know.