The Access Problem for Holes

[Hello to anyone out there!

Project: exposure therapy for my fear of posting sloppy/confusing stuff to this blog continues. So uh... sorry. I hope to return to regularly posting polished content soon.]

One of the least popular aspects of my philosophy of mathematics  is what I call Moderate Quantifier Variance. So I thought I'd explain one reason why I'm such a big fan of this view (and why I think you should be too) which has nothing to do with philosophy of mathematics.


The Access Problem for Holes

Consider our knowledge of holes. As discussed in D and S Lewis' On Holes, it's appealing take holes to be distinct from things like the air inside them, portions of the matter that `hosts' (e.g. portions of cheese with some diameter aronud a hole in Swiss cheese).

We seem to know how deeply a piece of cheese must be indented in order for there to count as being a hole in that piece of cheese. But (to a crazy philosopher, for a second) this knowledge can seem rather mysterious. For our senses tell us how the cheese is distributed through space. But how do we know where to draw the line: re how shallow a hole can be? No sensory experience seems to point out a single metaphysically special place to draw the line. So how can one explain the match between human beliefs about how shallow a hole can be and objective matters of facts?

It's appealing to say: there's no mystery about human accuracy here because if we had used `hole' differently (by taking the minimum hole angle to be larger or smaller) then the meanings of our words would have been different so these alternative hole-identifying practices would have yeilded true utterances. But since variant practices of "hole" individuation can require change in sentences whcih don't even use the word hole  (e.g. a purely logical sentence like the Fregean paraphrase for `there are >3000 things' can go from true to false). So it seems plausible that allowing such changes in meaning requires allowing variant meanings for logical vocabulary like ``there is'' as well as in the meaning of  ``hole''.

So I think a great way to solve this `access problem for holes'' is to accept

Moderate Quantifier Variance: there are multiple existential quantifier like meanings which the words `there is' can take on in different ideolects, (say English_1910, English_2010 etc.), though within any particular ideolect `there is' is univocal.

• These meanings are `quantifier like' in the sense that they obey the all instances of standard first order logical inferential rules/axioms for `exists' (within the language they belong to). 
• If there is a single maximally natural quantifier sense (as e.g. Sider thinks there is corresponding to our talk about `what there is' takes on when doing ontology), these variant quantifier senses need not be mere quantifier restrictions of this fundamental sense. 

I like to use Moderate Quantifier Variance to explain/vindicate the freedom mathematicians take themselves to have to introduce new logically coherent structures for study, by saying that when mathematicians consistently extend the axioms of pure mathematics by adding axioms describing some new mathematical structures like the complex numbers, their words can both give meaning to expressions like `complex number' and also change the meaning of `there is', so that these axioms express truths.

But whatever you say about math, I think moderate quantifier variance is already useful for understanding our knowledge of swiss cheese!

Trivializing Benacerraf and Monstrous Moonshine

[note: Sorry that this post is a little wordy. I'm trying to get back into blogging, and forcing myself to post stuff I'm not quite happy with is part of that. Also see this paper for a way more detailed and less aggro take on this issue.]

Remember that the access problem (aka the Benacerraf problem) for realists goes something like this: If moral/mathematical/etc realism is true, how can human accuracy and reliability about moral/mathematical facts be anything but a miracle or a mystery?

People like Justin Clarke-Doane and perhaps David Enoch, (call them The Trivializers) have been suggesting that we can answer all legitimate access worries besetting for realists about necessary domains like mathematics morals etc. just by “stapling together” two things to explain why we couldn't have easily been wrong (and providing a similarly trivial explanation for the sensitivity of our I won't discuss here):

• an  explanation (e.g.historical or evolutionary) for why we reliably believe certain moral/mathematical/etc claim,
• the fact that these claims are necessary truths

So, for example, a classic platonist might try answer access worries by explaining human reliability about mathematics like so:

TRIV: Mathematicians reliably believe truths because they reliably believe only those mathematical claims which can be proved in a certain formal system (e.g. ZFC) and this formal system is (necessarily) truth preserving w.r.t. the platonic mathematical objects.

Such an explanation is not likely to satisfy anyone who feels an intuitive access worry. For TRIV just explains one intuitively mysterious match between human psychology and objective which intuitively “cries out for explanation” (our acceptance of theorems that match whats going on in Plato's heaven) by positing another such mysterious match (our acceptance of axioms that match what's going on in Plato's heaven).

But it does (in some sense) suffice to explain the safety of human beliefs by deriving our reliability about realist facts concerning the relevant domain (i.e., that in all close possible worlds, our beliefs about these domains match up with the truth) from more general premises which the realist (if not their deflationary opponents) accepts.

And I take Trivializers to be suggesting that our intuitions that certain regularities involving necessary truths "cry out for" explanation in a deeper/more unified sense, which mere deductions of reliability/safety/sensitivity like TRIV need not provide, is an illusion.

There are two reasons why I don't buy this.

First, conceptually analyzing anything (from tablehood to justice) is infamously hard, but there are plenty of good paradigms for how to think about ”crying out for explanation” which seem to apply equally well to necessary and contingent regularities (e.g., norms that say we should preferring theories that have fewer degrees of freedom and or Kitcher’s idea of scientific explanation as unification).

Much more importantly though, embracing the Trivializers' position seems to have deeply implausible revisionary consequences for mathematical practice, since mathematicians sure seem to think that some regularities involving necessary truth truths can cry out for (non-trivial) explanation.

For example, consider this quote from a popularizing article about the history of John Conway's `Monstrous Moonshine' conjecture.

``Strangely enough, [the j- function]’s first important coefficient is 196,884, which McKay instantly recognized as the sum of the monster’s first two special dimensions.
Most mathematicians dismissed the finding as a fluke, since there was no reason to expect the monster and the j-function to be even remotely related. However, the connection caught the attention of John Thompson, a Fields medalist now at the University of Florida in Gainesville, who made an additional discovery. The j-function’s second coefficient, 21,493,760, is the sum of the first three special dimensions of the monster: 1 + 196,883 + 21,296,876. It seemed as if the j-function was somehow controlling the structure of the elusive monster group.'’’

Note that mathematicians already had separate proofs of facts about the j-function and monster group (and hence an `explanation' of for the match between these facts of the kind which TRIV provides, i.e., a deduction, from more general premises, of the fact that this relationship holds in all close possible worlds). But (once this match between apparently unrelated domains proved striking enough) they expected to find some further/deeper unifying explanation and this expectation guided choices for further research.

So I'm prima facie pretty suspicious of the idea that felt intuitive demands for unifying/satisfying/non-trivial explanations of regularities involving necessary truths are generally misguided. But maybe I'm not being charitable to Clarke Doane and/or he can find some way of separating the "this regularity cries out for further explanation" intuitions he wants to dismiss as unreliable from those which obviously do good work in mathematics.

Posthumus Vindication and Newton's Concept of the Derivative

In a recent Mind paper, `Incomplete Understanding of Concepts: the Case of the Derivative', Sheldon Smith vividly sets up some classic questions about Newton's concept of the derivative, and how later mathematical work can be seen as vindicating Newton.

However I'm not entirely convinced by Smith's answers to these questions.

Historical Background:

[Smith tells us how] Newton and Leibnitz had certain limited beliefs about the derivative

• that it was "the local rate of change of a function given by the slope of the tangent" so the derivative of x^2 kinda should be 2x
• that it was the limit as i goes to 0 of (f(x+i)-f(x))/i, hence derivative of x^2 was [(x+i)^2-x^2]/i which they thought was =(2xi+i^2)/i=2x+i=2x

but they did not have a very solid justification for the later reasoning (particularly the presumption that one can divide by i in the claim above).

Since then, mathematicians have defined multiple derivative-like notions which all let one defend reasoning like the above more rigorously, but don't always agree:

• the usual: the derivative of f(x) is the function f'(x) such that for every epsilon there is an i such that |(f(x+i)-f(x))/i - f'(x)| < epsilon
• the symmetric derivative: [like the normal definition but with (f(x+i)-f(x-i))/2i in place of f(x+i)-f(x))/i] (note that when f(x)= |x|, the symmetric derivative is 0 whereas the standard definition is undefined).
• a definition using infinitesimals
• a definition which also can apply to generalized functions like the Dirac delta function

Furthermore there is a common intuition that, in providing some of the definitions above and proving things with them, mathematicians like Weierstrass "justified [Newton's and Leibnitz's] thoughts" and that Newton and Leibnitz would have felt "vindicated" by subsequent developments of the derivative.

The questions:

Now, Smith argues that Newton didn't seem to be using any particular one of these modern concepts of the derivative.

• Newton didn't (somehow) implicitly have any of these precise concepts in mind, and which definition of limit he would have preferred to adopt (if he had been told about all of them) might vary with which one he found out about first.
•  There's no single "best sharpening" of what Newton believed/had in mind which must be accepted in limit of ideal science. We just have separate notions of derivative, each of which is mathematically legitimate. Thus we can't say that Newton meant, say, the standard contemporary notion of the derivative because he was conceptually deferring to the results of ideal science.

So he asks:

1. How `` should [one] think about the derivative concepts with which Newton and Leibniz thought''? 
2.  How ``could [Weierstrass] have managed to justify their thoughts even if their thoughts did not involve the same derivative concept as Weierstrass’s''?

Smith's Answers:
I take Smith's answers to the above questions to be as follows:

Q1: What was Newton's concept of the derivative [specifically, how does it effect the truth conditions for sentences]?

A: Newton's concept of the derivative (call it derivative_N) "only has a definite referent" in cases where all acceptable sharpening definitions of his concept agree.  So, for example, if the symmetric derivative and the standard derivative were both acceptable sharpenings, then expressions like `the derivative_N of f(x)=|x|' would fail to refer [or, perhaps, would refer to function which is undefined at 0 so that 'the derivative_N of f at 0' would fail to refer].

Q2: How was Weierstrass able to vindicate Newton, given that his concept of the derivative was different from Newton's?

A: One can vindicate Newton by justifying particular claims Newton made (e.g., about the derivative of x^2). And one can do this giving a proof of the corresponding claim employing Weierstrass's definition, if it also happens to be the case that all other permissible sharpenings of Newton's notion of the derivative would agree on this claim.

A Small Objection: 

I'm not entirely convinced by Smith's account of Newton's concept (Q1) for various reasons. But even if Smith is right about Q1, I think his answer to the vindication question (Q2) is fairly unsatisfying.

For suppose (as Smith seems to presume) Weierstrass vindicated Newton by showing the truth of particular claims he made about calculous -- that, say, what he expressed by saying ``the derivative of x^2 is 2x'' was true. If (as Smith's account of Newton's concept seems to tell us) the truth of this claim requires that all acceptable precifications agree in making ``the derivative of x^2 is 2x'' come out true, how can one adequately justify Newton's claim merely by discovering *one* such precificiation and showing that *it* makes the above sentence come out true?

A fix?

Maybe Smith could solve this problem (while keeping his account of the concept and the, IMO, good idea that vindicating Newton doesn't require assessing all possible derivative-like notions) as follows.

Say that "vindicating Newton's thought"  in the sense we normally care about (the in the sense that seems to have happened, and that, plausibly, Newton and Weierstrass would have cared about) doesn't require showing that some of Newton's specific mathematical utterances expressed truths. Instead, one can do it just  by showing Newton was right to believe some more holistic meta claim like `There is some mathematical notion which makes [insert big collection of collection of core calculous claims and inference methods] all come out true/reliable'.

Three Projects Involving Dispensing With Mathematical Objects

One of the many unfortunate things about academic fashions is that when a popular project goes out of fashion, superficially similar-looking projects which don't face the same difficulties can be tarred with the same brush.

Many people (myself included) feel cautious pessimism about formulating a satisfying nominalist paraphrase of contemporary scientific theories [one major issue is how to make formulate something like probability claims without invoking abstract-seeming events or propositions]. But we wouldn't want to over generalize.

 In this post I'm going to suggest three different motives for seeking to systematically paraphrase our best scientific theories (and many other true and false ordinary claims) in a way that dispenses with quantification over mathematical objects, and note that the requirements for success in the first (ex-fashionable) project are notably laxer in some ways than the requirements for the others.

[Note: I don't mean to endorse any of the projects below, but I think that 3 and Burali-Forti based versions of 2 are at least interesting.]

Three Motivations for Paraphrasing Mathematical Objects Out of Physics

1. General rejection of abstracta: You deny the existence of mathematical objects because you think allowing any abstract objects are bad. (this is the classic motive)
2. Explaining special features of mathematical practice by rejection of mathematical objects: You deny the existence of mathematical objects because you think that not taking mathematical existence claims at face value is allows for the best account of certain special features of pure mathematical practice, (e.g., by mathematicians’ apparent freedom to choose what objects to talk in terms of/disinterest in mathematical questions that don’t effect interpretability strength, or by the Burali-Forti paradoxes in higher set theory) not to take apparent quantification over mathematical objects at face value. 
3. Grounding math in logic/bringing out a claimed special relationship between math and logic: You may allow the existence of mathematical objects, but you’re moved by the close relationship between an intuitive modal notion of coherence/semantic consistency/logical possibility and pure mathematics to seek some kind of shared grounding and think that the coherence/logical possibility notion looks to be the more fundamental. As a result, it seems promising to seek a kind of "factoring" story, which systematically grounds all pure mathematics in facts about logical possibility, and all applied mathematics in some combo of logical possibility and intuitively non-mathematical facts.

 Distinguishing these motivations/projects matters, because what you are trying to do influences what ingredients are kosher for use in your paraphrases: 

If you have the first motivation (defending general nominalism), you need to avoid quantifying over any other abstracta, including: platonic objects called masses or lengths which don’t come with any automatic relationship to numbers, corporations, marriages, sentences in natural languages, metaphysically possible worlds, and (perhaps) propensities. 

But if you have the second motivation (defending mathematical-nominalism-adopted-to-explain-special-features-of-mathematical-practice), then quantification over abstracta which don’t have the relevant special feature, (e.g., objects in domains where we don't think appear to have massive freedom to choose what objects to talk in terms of or objects which don't give rise to a version of the burali-forti paradox), is fine. 

And if you have the third motivation, then quantification over any objects which aren’t intuitively mathematical-- or aren't mathematical in whatever way you are claiming requires a special relationship to coherence/semantic consistency/logical possibility -- is OK.

Pictureability and Definiteness in Mathematics

A fairly popular position in philosophy of mathematics is the following: there are definite right answers to questions about arithmetic, but not to questions about set theory (especially not to questions like the Continnum Hypothesis whose answers can be changed by forcing), because we have a clear notion of what the numbers are supposed to be like but not what the sets are supposed to be like.

This claim about clarity is sometimes supported by appeal to notions of mental picturing. The idea would be that we can picture all the numbers, but that we can't picture things like `all the subsets' of the numbers, or the hierarchy of sets as an item that contains, at each level, all subsets of the levels below. But I want to ask: In what sense can we picture all the numbers? In what sense CANT we picture the sets?

All the ways I can think of of picturing all the numbers involve some kind of "..." e.g. you might picture all the numbers by having a mental picture like the physical picture I have drawn below:

| || ||| |||| ....

When we use this picture to conceive of the structure of the numbers, we are employing a highly non-trivial method of picturing, which lets us represent the state of there being a countable infinity of numbers by considering some finite collection of drawings like the above.

And people certainly do *in some sense* draw pictures of the hierarchy of sets. At the beginning of a set theory class (and at the beginning of my college analysis class) the professor will frequently draw up a V on the board. In a way that is a picture of the heirarchy of sets. And, presumably, one could mentally picture the heirarchy of sets in the same.

For this reason I am (a little) skeptical about justifying the claim that we have a definite idea about the sets but not the numbers by appeal to considerations about what we can mentally picture. We can picture the numbers if we are willing to let "..." represent the idea of all finite successors to a stroke symbol. We can picture the hierarchy of sets if we are willing to let the increasing girth of the V represent the idea that each level of the hierarchy of sets contains all possible subsets of the level below. What (mathematical) objects one can picture depends on what methods of picturing we will accept, as allowing one to genuinely conceive of a situation by entertaining a mental picture.

At the very least, I think that if someone wants to argue that claims about numbers are definite and claims about sets are not on the basis of appeals to mental pictures, they owe us some kind of story or explanation for why one method of picturing is OK and the other is not.

Angst Over the Ordinals

Ohhh, which of these three options is correct? Given my focus on philosophy of math it's mildly embarrassing not to have a fixed position on this, but I keep going back and forth...

1. Just say the hierarchy of sets goes "all the way up"

2. Say the hierarchy of sets goes "all the way up" in the sense that it contains ordinals corresponding to every distinct combinatorially possible way for some objects to be well ordered *except for the one that it, itself is an instance of*. (this would be appealing but i think it may be impossible to spell out in a consistent way)

3. Say that the hierarchy of sets goes up at least far enough to satisfy the axiom of infinity+ the rest of ZF, and leaves it vague what there is beyond the inacessables - much as our concept mountain leaves it vague how many really tiny mountains there are given that there is such and such a bit of lumpy terrain.

Five Reasons to be a Modal Carnapian

I currently believe mathematics is best understood in terms of combinatorial possibility plus Carnapian framework stipulations. One reason for thinking this is, of course, that understanding mathematical objects in this way lets you tell a nice story about access to abstract mathematical objects, like the one I tell in my dissertation! But here are 4 other reasons.


1. Thinking in terms of combinatoiral possibility lets you solve the bad company objection for neo-carnapianism about objects.


You can say: It is OK to stipulate that there are sets but not to make the stipulations for tonk, or for Boolos parities because...

One can safely extend one's language by introducing object stipulations S if and only if it would be combinatorially possible for the objects one's language currently acknowledges at each possible world to be supplimented by new objects in such a way as to satisfy S.

2. Thinking about things in terms of combinatorial possibility + carnapian framework stipulations explains why certain mathematical questions are substantive questions while others are not (substantive ones involve disagreement over combinatorial possibility as well as over which from among the many objects which it would be combinatorially possible to stipulate you actually do stipulate)

For example:

There is no substantive mathematical question about whether there are sets or categories or both, because everyone agrees that it would be combinatorially for there to be objects satisfy Carnapian framework stipulations phrased in terms of combinatorial possibility for the sets, and for the categories.

There is a substantive mathematical question about whether the Goldbach conjecture is true, because it is combinatorially impossible that the Fs and the Gs should satisfy the informal (non-first order logical!) framework stipulations for sets could yield different answers to Goldbach.


3. "Mathematical Hypothesis"-based alternatives like structuralism, fictionalism and plenetudinous platonism already need *some* powerful modal notion or a sense of possible and impossible which can take into account vocabulary that goes beyond the connectives of first order logic, if they want to capture intuitive claims about all questions in arithmetic having right answers.

Intuitively there are right answers to at least all questions about arithmetic. Each first order logical hypothesis about the numbers necessitates right answers to all questions about arithmetic. Therefore, any mathematical hypothesis that claims to cash out the truth conditions for our claim that phi in terms of the claim that if H then phi must crucially use non-first order vocabulary. Therefore if you want to make some hypothetical view of math work, you must either bite the bullet that some simple questions of arithmetic have no answer, or invoke a more mathematically powerful notion of possibilty/consequence

4. The notion of combinatorial possibility is very crisp and simple, indeed it maybe well the crispest and simplest notion that yields enough mathematical power to reconstruct intuitive verdics about truthvalues in arithmetic as per (3).

For example, if you accept Tarski's definition of logical truth in terms of facts about how it would be possible to reinterpret all properties and relations involved (where possible doesn't mean that a person could give a rule for how to do it, or somehow list the new extensions) then you pretty much already accept the powerful notion of how it would be in principle possible for an arbitrary n-place relation to apply to some objects which I have in mind when I talk about combinatorial possibility. So if you're not too much of a finitist/intuitionist to accept the very broad sense of in principle possible reassignment of extensions to predicates used in the Tarski definition, I think you should be OK with my notion of combinatorial possibility.

Secondly, combinatorial possibility faces none of the quandires that make people skeptical about the notion of metaphysical possibility e.g. it is possible for something to be both red all over and green all over, or for there to be zombies? We avoid these problem because facts about combinatorial possibility ignore all metaphysical facts about particular properties and relations involved. In some cases one might perhaps argue that some natural language sentences are vague with regard to what pattern of relationships between objects they assert. But once you specify that you are asking whether e.g. combiposs( Ex Redallover(x)&Greenallover(x)), it is perfectly clear that this is combinatorially possible, since it would be combinatorially possible to choose extensions for Redallover() and Greenallover() that make this true.

What is Combinatorial Possibility?

At the moment I think mathematics is best understood in terms of neo-carnapian/neo-logicist existence conditions for mathematical objects plus a kind of specifically mathematical modality (along the lines considered by Charles Parsons) which is looser than metaphysical possibility, and which I call "combinatorial possibility".

Here's a new way I thought of to explain what I mean by combinatorial possibility:

Combinatorial possibility is just like Tarski's notion of logical possibility, except a) we drop the assumption that you have to shanghai objects from the actual world to use in making a given claim true, allowing arbitrary choices of domain as well as arbitrary reassignments of extensions to properties and b) we allow more vocabulary to count as "logical vocabulary" in the sense that when reassigning extensions to predicates you can't change its meaning.

What more vocabulary?

In general anything like "finitely many" or "equinumerous" which functions like logical vocabulary in a sentence that its effect on the truthvalue of the whole sentence is systematically determined just by the domain and extension of relations in a set theoretic model is OK. Call these semi-logical expressions.

However I conjecture that we can cash out all of these all the semi-logical expressions, or at least all the ones that occur in modern mathematics and and its applications by merely using two things:
- an actuality opporator @F, so that you can say things about how it would be combinatorially possible for the things that are actually kittens, to be related by liking to the things that are actually baskets, e.g. there are enough different kittens and few enough baskets that it is combinatorially impossible that each kitten slept in a different basket last night.
- nesting, so that you can ask questions about the combinatorial possibility or impossibility of questions which are themselves described in terms of how it would be combinatorially possible to supplement them e.g. There are people at this party representing all combinatorially possible choices of whether to take sugar and/or milk in your tea = It would be combinatorially impossible to supplement the actual people with some additional person who differs from all actual people with respect to either whether they take coffee or whether they take tea = not combiposs(Ex person(x) & Ay [if @person(y) then ~x=y , and takessugar(x)iff~takessugar(y) and takesmilk(x) iff~takesmilk(y)) ] *


*yes, when you have multiple nesting of claims about combinatorial possibility, actuality opporators will need to be indexed to a particular instance "combiposs" so you will have combiposs_i and @F_i.

Dilemma re: (Platonist) Structuralism

I've got to go reread my Shapiro. But before his smooth writing bewitches me, let me note down the very simple objection that I am currently unable to see how he would answer.

Structuralism is traditionally motivated by the desires to address a problem from Benacerraf: that there are multiple equally good ways of interpreting talk of numbers as referring to sets, so that either answer to "what set is the number 3" seems unprincipled. But now:

If you are not OK with plentiful abstract objects, you can't believe there are abstracta called structures.

If you are OK with plentiful abstract objects, then you can address this worry by just saying that the numbers and sets are different items. Certain mathematics textbooks find it useful to speak as though 3 were literally identical to some set, but this is just a kind of "abuse of notation" motivated by the fact that we can see in advance that any facts about the numbers will carry over in a suitable way to facts about the relevant collection of sets named in honor of those numbers. One might argue that analogous abuse of notation happens all the time in math e.g. writing a function that applies to Fs where you really mean the corresponding function that applies to equivalence classes of the Fs. This route seems like a much less radical move than claiming that basic laws about identity fail to apply to positions in a structure e.g. there is no fact of the matter about whether positions in two distinct structures (like the numbers and the sets) are identical.

Invention, Discovery and Creativity in Mathematics

Non-philosophers I meet sometimes ask: do I think mathematical facts are invented or discovered? IMO, this is a weird question - and not one that comes up much in the phil math literature- because the contrast between invention and discovery is not very well defined. For example, did Alexander Gram Bell *invent* the telephone, or did he *discover* that putting components together in a certain way would build a telephone? Intuitively, one might say both.

Maybe what people mean to be asking by this question is just this: do mathematicians bring new mathematical objects into existence, or do they discover already existing objects? For, paradigmatic cases of invention typically do involve creating a new physical object, while paradigmatic cases of discovery involves visiting an already existing physical object. So e.g. Columbus discovered America (because it already existed and he went to visit it) whereas Bell invented the telephone, by physically creating the first prototype.

However, the contrast between invention and discovery can't really just track the distinction between cases where a new object is made vs. not. This is because making a new thing isn't required for invention *or* discovery. Consider an imaginary scenario where Bell just thought up a plan for a telephone, and told someone else who physically constructed the first one years later. Bell would still have invented at telephone, if he though up the plan and then worked out from known principles that the plan would work, but never made one.

While we are talking about invention and discovery, I think there's a third notion -artistic creation (e.g. what happens when someone composes a story or a poem)- which bears an interesting relationship to mathematical discovery. When a writer writes a story, they are putting down a sequences of sentences which already exists as an abstract object.

I mean, suppose that the story teller composes a story today. If a linguist said yesterday 'no intelligible sequence of English sentences has property P', the and the sequence or sentence which the story teller writes down today has property P, then then the linguist's claim yesterday was false. The domain of potential counterexamples to linguistics claims today, already contains all sequences of English sentences which literary ingenuity could ever devise. Note also that to compose a story or poem doesn't require writing it down anywhere, (the person in the Borges story who has time stop so he can finish writing a poem before he gets shot still counts as creating the poem). For this reason the task of literary "creation" doesn't really seem to involve creating anything, (neither a physical artifact, nor an abstract string of sentences), but rather directing your attention to an abstract object that already exists - carefully sorting out which string of sentences will combine certain varied and subtle properties in the right way.

Now, if I'm right about this- the creativity of a poet or novelist doesn't need to involve creating any new object, but rather amounts to discovering a pre-existing string of sentences which has a certain property - this suggests a potential confusion about the relationship between mathematical creativity and ontology. Arguably, mathematical creativity is much like literary creativity. But, if mathematical creativity is like literary creativity, it does not follow from this that the mathematician creates the mathematical objects he describes, or that he creates anything else. For (if the above is right) literary creativity isn't a matter of bringing new objects into being, but rather a matter of discovering, amid the combinatorial explosion of possible sequences of English sentences, one that has a certain special features.

Why Math and Morals Aren't Companions in Guilt

Intuitively, many people feel that epistemic worries about moral facts (if there are moral facts, how to explain why our moral intuitions should be even even remotely correct about them?) are WAY more serious than epistemic worries about mathematical facts (if there are mathematical facts, how to explain why our mathematical intuitions should be even even remotely correct about them?). But is there really a difference here?

Well, here's one thing that I think does make a difference: mathematical claims about number theory have direct and specific consequences for stuff that we can check by logic and/or scientific observation.

-what will happens whenever a person or a computer to successfully applies a certain syntactic alogorithm
-how many apples-or-oranges do you have when you have n apples and m oranges (cf Frege for why this is a logical fact)

This matters because, plausibly, the need to get these concrete applications right likely prevents our beliefs about number theory from getting too off the wall - whereas, our moral intuitions have no such multitude of consequences which are directly checkable by logic and observation.

Epistemology verses Foundations in Philosophy of Math

The epistemology of math task: Get a true theory of what under what circumstances a person counts as knowing something. Or, at least, square our beliefs about what people have or lack knowledge of what particular mathematical beliefs, with general beliefs about what’s required for knowledge (e.g. causal contact.

The foundations of math task: extend our mathematical knowledge.

I claim that making this distinction matters a lot, because:

Arguments that are helpful for foundations of math are (in themselves) useless for the epistemology task. Suppose we have a working derivation D of certain facts of arithmetic from logic. And suppose we have a perfectly adequate, intuition-matching story about what it takes to count as knowing the relevant logical facts.
This still does not allow us to account for current knowledge of arithmetic (i.e. reconcile our theory of knowledge with the intuition that people now know things about arithmetic). This is because - in general - it is not enough for S to know that P, for P to be true, S to believe that P, and P to be derive*able* from things which S knows. In general, the subject S needs to have some kind of access to the derivation. The mere fact that I believe that P, and P can be proved from other things that I know, hardly suffices to establish that what I have counts as knowledge. If a lawyer is asked to show that some contractor knew that a bridge was safe, it doesn’t suffice to show that one *could* derive from laws of physics and facts about the blueprint which the contractor knew that the bridge was safe - we also need to suppose that the contractor did derive it, or get testimony from someone who derived it or the like.

Hence, a foundational argument which derives (say) one body of mathematics from premises that are more certain is not directly relevant to the general epistemological project.

Conversely, an accurate epistemology of mathematics can be almost perfectly useless to the task of setting some shaky region of mathematical theory on firmer foundations. For example, one classic account of knowledge is reliablism. If we modify reliablism so as to apply non-trivially to mathematics (following suggestions by Linnebo and Field) we get the idea that someone has knowledge if they have a true belief which is reliable in the sense that: they accept a sentence which expresses p, and if that sentence had not expressed a truth, they would not have accepted it. This is a perfectly decent candidate for a general account of mathematical knowledge. But note that, even supposing that it is right, it does nothing to help satisfy foundational desires for, say, more secure foundation for the axiom of choice. If someone has foundational worries about the axiom of choice, they have worries about whether it is true. They might express these worries by saying ‘how do you know that the axiom of choice holds?’ but the emphasis here is on truth, not on knowledge. It would be silly to respond by saying that we know AC because AC is true, and we have reliable beliefs (as defined above) to that effect. What the foundation-seeker really wants is to know whether AC. They want to acquire knowledge about whether AC, not get a general theory of what it would take to count as knowing AC.

So, I have been trying to argue that it’s important to make a distinction between the epistemological project of trying to come up with a general theory of when someone knows something about math, and the foundational project of trying to make it the case that we know more things about math, by supplementing inadequate arguments with additional arguments that appeal to premises which are already known. The one focuses on the most bland an uncontroversial cases of mathematical knowledge, and tries to reconcile our other beliefs about the nature of knowledge with our particular judgments about this case. The other seeks out the most controversial regions of mathematical claims, and seeks to secure knowledge for us about these claims, by connecting them to claims that are more securely known. Enticing answers to one project can easily seem to frustratingly miss the point for someone who is interested in the other, as shown in the examples above. Hence it’s important to make the distinction.

However, this is not to say that there’s no relationship between the epistemological and foundational projects. Thinking about big picture issues about justification in general, can influence your judgments about particular cases. A kind of trivial example of this is intuitions about what you can take for granted, while still counting as being justified. Just off the top of one’s head, it can seem attractive to say that someone doesn’t count as knowing that P if all they can give is a circular justification for P, an infinite regress of justifications, or a justification that comes to a halt at a certain point. But when you consider these three options together and notice that they exhaust all the possibilities, you will likely be inclined to give up the principle that someone who can only give a justification of one of these kinds must thereby not count as having knowledge. So, if two realists about AC are attempting to provide and evaluate firmer foundations for AC, it may be helpful for them to general questions about what’s required for knowledge and justification – to make sure that their evaluation of the evidence in this case, doesn’t depend on assumptions about justification which turn out to be incoherent or conflict with what they take to be sufficient evidence more generally.

Are mathematical truths "substantive"?

One thing that that has caused me great puzzlement (in the past few years), is the question of whether math tells us anything 'substantive'. I want to suggest that our intuitive notion of "substantiveness" combines two distinct notions, which come apart in this case.

- mathematical truths DONT rule out any physically or even metaphysically possible states of the world. (This is just another way of putting the truism that mathematical truths are necessary, hence compatible with every metaphysically possible world. I like putting things this way, because it doesn't suggest that necessary mathematical truths arise from something (mathematical objects?) causally blocking any person that tries to being both more than three feet long and less than two feet long)

- mathematical truths DO combine with our background beliefs to lead us to form expectations we wouldn't have formed otherwise (e,g. about the results of future counting procedures, about the programs)

Presumably you admit that these are at least nominally different properties. But you might still wonder *how* these two things could come apart. How could knowing any proposition be useful, if this proposition didn't rule out any possible states of the world? Here's what I think the answer to that is in a nutshell:

Some mathematical facts (i.e. facts which are derivable from math and logic alone) which are useful because they tell us that whenever one description of the world holds, then so does another (e.g. anything that accelerates from standstill at this rate for this amount of time travels that distance, anything that's less than two feet long isn't three feed long.)

And here's the answer in more detail.

FOL as the language for science

Maybe I'm missing something here...

Quine suggests that we adopt first order logic as the language for science. But, first order logic can't capture the notion of 'finitely many Fs'. It can only express the claim that there are n Fs for some particular n. Yet, we do understand the notion of finite, and use it in reasoning (e.g. if there are finitely many people at Alice's party, there is one person such that no one is taller than him) and potentially in science. Hence, we should not adopt first order logic as the language for science.

[The standard way to try to get around this, is by talking about relations to abstract objects like the numbers (There are finitely many Fs if there's a 1-1 map from the set of things that are F to the some set theoretic surrogate for the numbers). This would give you the right extension, if your scientific hypothesis could say that something had the structure of the numbers. But first order logic can only state axioms, like PA which don't completely pin down the structure of the numbers. Any first order axioms which you use to characterize the numbers will have non-standard models. This is Putnam's point in his celebrated model theoretic argument against realism. So, if you take this strategy, rather than saying that there are finitely many people at Alice's party, you can only say that the number of people is equinumerous items that satisfy a certain collection of first order axioms. And this does not rule out non-standard models.]

Is Math Logic?

Is mathematics just a branch of logic? This is the first question many people ask about philosophy of math (sometimes with a vague idea that a) it would solve some kind of metaphysical or epistemological problems if math were logic or b) it's been proved that math isn't logic). Well, unsurprisingly, the answer depends on what you mean by 'logic'. Here are some different senses of the word 'logic' that one might have in mind.

1. first order logic
2. fully general principles of good reasoning
3. a collection of fully general principles which a person could in principle learn all of, and apply
4. principles of good reasoning that aren't ontologically committal
5. principles of good reasoning that no sane person could doubt

The sense in which it has been proved that math isn't logic is (to put things as briefly as possible) this: You can't program a computer to spit out all and only the truths of number theory.

This fact directly tells us that the mathematical truths are not all logical truths, if we understand "logic" in sense #1 - since we *can* program a computer to list off all the truths of first order logic. And it also tells us that the mathematical truths aren't all logical truths in sense #3 or #5 either - if we are willing to make the plausible assumption that human reasoning can be well modeled in this respect by some computer program. For if all human reasoning can be captured by a program, then so can all human reasoning from some starting finite collection of humanly applicable principles, and so can the portion of human reasoning that no sane person could doubt (to the extent that this is well defined).

However, if by "logic" you just mean #2 -fully general principles of reasoning that would be generally valid (whether or not one could pack all of these principles into some finite human brian)- then we have no reason to think that math isn't logic. We expect the kinds of logical and inductive reasoning we use in number theory (e.g. mathematical induction) to work for other things (especially for things like time, which we take to have the same structure as the numbers). If Jim didn't have a bike on day 1, and if, for each subsequent day he could only get a bike if he had already had a bike on the previous day, then Jim never gets a bike. If there are finitely many people at Jane's party, there is one person such that no one is taller than them. The laws of addition are the same whether you are counting gingerbread men and lemon bars, or primes and composite numbers. And this doesn't just apply to principles of mathematical reasoning which we actually accept. We also expect any *unknown* truths about the numbers (as the smallest collection containing 0 and closed under a transitive, antisymmetric relation like successor) to be mirrored by corresponding truths about any other collection of objects which contain some other starter element and are as few as possible while being closed under a transitive, antisymmetric relation (be this a collection of infinitely many rocks, or a collection of some other abstracta like the range of possible strings containing only the letter "A"). Hence, it is plausible that every sentence about numbers is an instance of a generally valid sentence form containing only worlds like "smallest", "collection", "antisymmetric" "finite" etc - and every mathematical truth is a logical truth in this regard.

Finally, if by "logic" you mean #4- ontologically *committal* good reasoning, the answer depends on a deep question in meta-ontology. For, it is well known that standard mathematics can be reduced to set theory, which in turn can be reduced to second order logic. But what are the ontological commitments of second order logic?

People have very different intuitions about whether we should say that there really are objects (call them sets with ur-elements or classes) corresponding to "EX" statements in second order logic. Does the claim that "Some of the people Jane invited to her party admire only each other, so if all and only these people accept, she will have a very smug party" assert the existence of objects called collections? More generally: the quantification over classes in second-order logic ontologically committal? Statements like the one above certainly seem to be meaningful. And, it turns out not to be possible to paraphrase away the mention of something like a set or class, in the sentence above, using only the tools of standard first order logic. This reveals a sense in which we treat reasoning about abstracta like classes (or, equivalently for these purposes, sets with ur-elements), very similarly to ordinary objects in our logical reasoning about them. But is this enough to show that second order logical is ontologically committal (and hence not logic at all, according to meaning #4)?

I propose that the key issue here concerns how closely ontology is tied to inferential role. Both advocates and deniers of abstract objects will agree that many of the same syntactic patterns of inference that are good for sentences containing "donkey" and sentences containing "set". But what exactly does this tell us about ontology? If you think about ontological questions as being questions about what the logical role of an expression in a given language, this tells you something very decisive. On the other hand, if you think about ontology can swing somewhat free of the inferential roles of sentences in languages (so an expression can have an object-like inferential role without naming an object), it's open to you in principle to say that - however similar their logical role- second order quantifiers are not ontologically committal. On this view, claims about sets with ur-elements are just ways to make very sophisticated claims (generally claims that could not otherwise be finitely expressed) "about" the behavior and relationship between ur-elements, and true claims about pure sets (i.e. sets that can be built up just from the empty set) are true in a way that does not involve any particular relationship to any objects, but can illuminate the necessary relationships between different expressions about classes that do have ur-elements. [At the moment I prefer the former view, that quantification in second order logic is ontologically commital, but this is a subtle issue]

Thus, to summarize, it is fully possible to say - even after Godel- that math is the study of "logic" in the sense of generally valid patterns of reasoning. However, if you say this, you must then admit that "logic" is not finitely axiomatizable, and there are logical truths which are not provable from the obvious via obvious steps (indeed, plausibly ones which we can never know about). Note that to make this claim one need not give up on the idea that logical arguments proceed from the obvious via obvious steps. For, if you take this route you can (and probably will want to) distinguish the human practice of giving logical arguments, from the collection of logical truths. You can say: only some of the logical truths seem obvious to us, and only some of the logically-truth-preserving inferences seem obviously compelling to us. We make logical arguments by putting these inferences together to get new results which are also logical truths. But (what Incompleteness shows) is that not all logical truths can be gotten from the ones that we know about. You can even claim that mathematical truths are logical in the further sense of not being ontologically committal, if you allow (contrary to the usual close association between objecthood and logical role) that the set quantifiers in second order logic are not ontologically committal.

Miniature Phil Math

Almost everyone agrees that our mathematical talk is practically helpful. Unlike astrology, doing math helps us build bridges. But how is math practically helpful? And does the way in which talking about numbers is practically helpful give us any reason to think numbers actually exist?

In this tiny essay I will propose a theory of how the practice of talking as if there were numbers is helpful. Then, I will say that we can appeal to numbers to explain how this practice is helpful, though there are also other correct explanations for this phenomenon which do not commit themselves to numbers. I will conclude by turning to the question of whether there are numbers. On the basis of the previous section I will propose that we do not *need* to posit the existence of numbers to explain the practical usefulness of our mathematical talk. However, we have another reason to believe in numbers which is the following: We want to make statements like "the number of cupcakes doubles every day" true (under certain circumstances), and the pattern of inferences we make with this sentences is quantificational. But this (being describable by some true sentences associated with a existential pattern of inferences) is the only thing that the many different kinds of non-mathematical objects which intuitively exist have in common.

1. How talking about abstracta like numbers is helpful

Talking about abstract objects, like numbers, is helpful because it lets us economically hypothesize patterns 'in the world around us' as well as patterns that might be described as artifacts of language (patterns in which distinct descriptions are logically or otherwise necessarily equivalent). We can say one sentence (about numbers) that will cause people to be willing to infer infinitely many different sentences that aren't about numbers.

For example, suppose I say: "The number of cupcakes doubles every day" This is a claim that quantifies over numbers and days, in the sense that we might represent it as "Ad An if d is a day, and n is a number, then there are n cupcakes on d there are 2n cupcakes on the day after d. "
Hearing this single sentence will lead my listeners to accept many different statements that do not quantify over cupcakes:
"if Ex7 cupcakes today Ex14 cupcakes tomorrow."
"if Ex8 cupcakes today Ex16 cupcakes tomorrow."
"if Ex7 cupcakes tomorrow Ex14 cupcakes the day after tomorrow."

2. What role do abstract objects play in explaining why talk of abstract objects is helpful?
Now we can ask: what role do various objects play in explaining the success of this talk? We might explain the helpfulness of my statement by saying that it is helpful because it...
- lets us track and predict what cupcakes there are and will be
- lets us track *the pattern in* what cupcakes there are and will be
- lets us track and predict how *the doubling function* relates *numbers*, and then predict what cupcakes there will be when, by relating this to facts about the behavior of the doubling function.

It seems to me that all of these are intuitively decent explanations. I take it that what we have here is a typical phenomenon where the same phenomenon (a war) can be explained by accounts that quantify over various different objects (countries vs. people vs. atoms). However, not much would be lost if we just stuck to giving the first explanation, which does not involve any mention of abstract objects.

3. Are there numbers? A good and bad reason for believing in numbers.
If this story about how math is practically helpful is right, should we believe that there really are objects of the kind talked about in these explanations e.g. patterns in the provenance in cupcakes, or numbers and a doubling function?

I don't think there is an *inference to the best explanation* for the existence of patterns in the provenance of cupcakes, or numbers from the helpfulness of this talk. It's not the case that we *need* to posit abstract objects called "patterns in the provenance of cupcakes" or "numbers" to explain how saying the thing described above could help people cope with the cupcakes around them.

Instead, I think it's reasonable to believe in numbers because we have an intuitively true sentence ("the number of cupcakes doubles every day") which allows a existential pattern of inferences - and playing this logical role is all there is to being an object.

The idea here is that when we look at the variety of different "objects" in the world e.g. electrons, magnetic fields goats, holes, waves, contracts, countries, these different kinds of talk don't seem to have much in common with regard to their relation to the physical world. What they do have in common is the pattern of inferences we make between sentences between them. In each case we accept sentences, such that the inferences with these sentences in are elegantly captured (in first order logic) by something of the form "Ex Fx". Now it turns out that talking about numbers and the doubling function shares this same feature.

Contrast w/ Tait "The Platonism of Mathematics"

Both my view (Lumpist Platonism) and Tait's might be considered unusual or quirky versions of platonism. Platonism (in phil math) is the view that mathematical objects exist.

I think that the world is fundamentally (something like) a space-time manifold [as opposed to a set of facts, or a set of objects and relations], and that all statements are true or false in virtue of how the manifold is. This includes statements about objects, and different statements about objects will correspond to very different claims about the state of the manifold (e.g. saying that there's a table vs that there's a whirlpool vs a trade deficit vs. a marriage contract vs. a number or string of symbols or a proposition). So facts obtain, and objects and relations exist, in virtue of how the physical stuff of the world is configured, not vice versa. Necessary truths (like all statements of pure math) correspond to the trivial claim about the state of the manifold (one that doesn't rule out any possible configurations).

Tait, as I understand him, thinks that mathematical sentences show that objects exist by constructing suitable objects. He writes "A proof is a presentation or construction of an object: A is true when there is an object of type A and we prove A by constructing such an object."

Both of these views contrast with what you might call a "two worlds" version of platonism. On this view: in addition to whatever objects exist in virtue of the physical stuff of the world comporting itself a certain way, there is also an "extra" component of reality. So far as I understand the force of the word "extra" here, the point of saying that there's an extra component of reality is this: An infinite and putatively exhaustive description of the world given purely in the language of microphysics e.g. (this point has that property, this point has that property etc.) would be missing out on the existence of sets, *in some stronger sense then the sense than in which it would be missing out on rabbits and trade deficits*.

Tait and I also agree that sentences are the right place to start when considering how semantics relates to metaphysics and ontology. For a sentence to be meaningful you just need the whole sentence to somehow make a claim about the world. Thinking about particular words in the sentence as having favored relations with particular chunks of matter will help in some cases but not others.

However, I disagree with Tait on some really important points:

Firstly, I don't really understand what he means by construction. The best sense I can make of the idea of constructing mathematical objects (how can you bring an abstract object into being?) is that it's something like the way I can create a) a marriage contract with another person by signing things the courthouse, or b) the set with ur-elements {Sharon's mullet} by giving myself an ill-judged haircut and thereby bringing a particular mullet-token into being, and hence it's corresponding singleton. But if this is what he had in mind, then...
a) it has the (at the very least) wildly counterintuitive to say that there wasn't a number between 3 and 5 before someone wrote down a proof inscription.
b) quantification in math works very weirdly and differently from quantification in general. For, since people have only written finitely many proofs there will be some number - say 347892-, such that no one has inscribed a proof of "3457892 has a sucessor". On the other hand, we certainly have inscribed proofs of "Ax if x is a natural number then x has a successor". So it would seem that the general statement is true. But the instance is (at the moment) false.

Secondly, Tait doesn't seem to allow that quantified statements of arithmetic (like, say, the Godel sentences for various formal systems) already have truth values now. He seems to think we are free to choose which kinds of proofs to construct (i.e. what formal system to adopt). And then he says that "the incompleteness of formal systems such as elementary number theory can be proved by induction, is best seen as an incompleteness with respect to what can be expressed in the system rather than with the rules of inference." And he points out that by extending the language (and adding suitable instances of the induction schema) you can prove the Godel sentence (and con) for this system.

But when I wonder about e.g. con(PA+X) [it's pretty hard to wonder about con(PA) imho] or con(ZFC), I'm not just wondering whether I could extend my formal system in such a way as to allow these sentences (or their negation) to be derived. Obviously, I could start making derivations (and hence constructing objects, for Tait) in any formal system I want. Nor am I pondering what kind of lifestyle choice to adopt in the future. Rather, I think that *right now*, I understand what it means to ask whether there's a proof of 0=1 from ZFC. And this is what I want to know. Is this sentence provable in that formal system or not? Is there such a proof or not?To the extent that we can ever be sure that we really understand something, and are asking a sharply meaningful question, this is it! [I think this may be why my advisor PK disagrees with Tait too]

Overall, I'm tempted to suspect that Tait is getting into bed with unattractive antirealism because he wants to avoid an epistemological problem. He sees how (/doesn't worry about how) you could know that something exists if you are able to bring it into existence (construct it). Such knowledge is sometimes called "maker's knowledge". And then he wants to say what mathematical knowledge is, in such a way that all mathematical knowledge turns out to be accessible in this way - which leads to weird consequences about large numbers, and unknown arithmetical facts.

In contrast, if you use the ...ahem... magic of Sharon's thesis, to provide a general naturalistic mechanism for how physical creatures could have gotten a faculty of reliable rational insight into abstract mathematical/logical truths :) , then you don't have to do any of this fancy (and potentially distorting) footwork.

Parsons and Intuitability

I've just been summarizing CH1 of Charles Parsons' Mathematical Thought and it's Objects. It set me thinking that Parsons is oddly concerned with whether you can "see"/percieve/intuit mathematical objects. I say oddly, because IMO what matters for assuaging worries about the weirdness of mathematical objects or the weirdness of our knowing about them (which seems to be part of his aim) isn't whether we can strictly speaking *see*/perceive/intuit abstracta, but rather a) whether positing abstracta isn't a violation of Occam's razor and b) how there can be enough of a connection between mathematical facts and our dispositions to form beliefs about them, for what we have to count as knowledge.

I mean: even in the empirical case, questions about what we can see, as opposed to merely inferring from what we see are super murky. Who knows whether you can "see" that the light is on vs. that the electricity is back on vs. that Jones succeeded at his task etc. as opposed to inferring them or justifiably and reliably forming true beliefs about these subjects)? What matters (for the epistemology worry b) is just that there needs to be some suitable and clear reliable mechanism at work leading you to form true beliefs on these subjects - as there obviously is in the empirical case of the light. Once we see how this reliable mechanism could work, it's (in my opinion) a matter of indifference whether you want to describe this mechanism as seeing the light and then immediately and unconsciously but justifiably inferring that the electricity is back on vs. directly seeing that the electricity is back on.

And the same goes for knowledge of mathematical objects. What we'd like is something that was like perception in the sense that it provided an unproblematic mechanism whereby we could get the relevant kind.Once we have that in place, we can say whatever we like about whether someone staring at a piece of paper can see/percieve/intuit that there's a proof of SS0+S0=SSS0 in PA, or a palindrome containing the word 'adam' vs. merely reliably and justifiably infer these statements from the concrete object that they do see. The million dollar question is how we manage to do this putative seeing/inferring correctly.

Similarly, if someone thinks that construing math as stating truths about genuine abstract objects is a violation of Occam's razor, (as per objection a) they aren't going to be impressed by claims to "see" the abstract object (a string) in the concrete object (a series of inkmarks). When the Platonist stares at the sheet of paper and says they are seeing that there's a proof SS0+S0=SSS0, the Fictionalst will say that they are seeing that there would have to be a proof in the relevant mathematical fiction, and the modalist will say you are seeing that a certain proof is possible.

My point here is not to knock Parson's interest in the relationship between concrete things you can see and abstract mathematical objects. Hearing him talk about this connection was a decisive inspiration for my own view, and I think it's absolutely crucial to think about the concrete physical processes going on when we form and revise mathematical beliefs, if you want to understand how creatures like us could know about (or even think about) something as abstract as math. But I would claim that the key point about string inscriptions isn't what they represent/allow us to intuit (can you stare through the string inscription to the string itself?, can you at least see that a certain string exists?), but (as it were) what you take these inscriptions to represent, i.e. how you are willing to form and revise your beliefs about other things, like strings as abstract objects, in response to seeing them. This is what starts to give us traction in linking up our dispositions to form mathematical beliefs to mathematical facts, to answer challenge (b). (IMO answering challenge (a) requires something else entirely, namely Lumpism, but more about that in the next post)

Parsons Mathematical Thought and its Objects CH1 summary

No one I've talked to is really sure what's going on. Especially me. But here's my current best guess. Maybe the magic powers of saying something wrong on the internet will help us work our way incrementally to a better interpretation.


1. Abstract objects defined + generic worries about them

Mathematical objects would be abstract objects = acausal, not located in space and time.
Worry: They aren't perceptable, if perceiving something requires locating it. Maybe this suggests there are no such things?
Response:
- electrons don't seem to be directly perceptable either, but they exist
- if we say that mathematical objects don't exist then we will have to explain why talking as if they did is so helpful for science.
- it's not clear whether we can avoid quantifying over abstract objects, hence (if we accept Quine's criterion) saying that they do exist.

2-3 What is an object?

It's hard to answer the question 'what is an object?' since unlike with gorillas we can't point out a contrast class of things that aren't objects.

the right answer: logical role
Philosophers usually ask 'what's an object?' in the context of trying to figure out how language can relate to the world - how we can talk about objects. For these purposes we can define being an object in terms of logical role: objects are what we talk about by using singular terms (e.g. 'Bob' in Bob is happy= Happy(Bob)) and quantification (e.g. 'Ex x happy').

other conceptions of objects/requirements philosophers have had for objects...

i. actuality/causal powers
Digression about Kant: general notion of object vs. "Wirklichkeit"
Kant invented the phrase 'concept of an object in general'. Kant's "categories" are concepts of an object in general. He is conflicted about whether these categories have to be perceivable by the senses [and hence whether "the concept of an object in general" would allow abstract objects?]
a) the categories are supposed to be derivable from logic and general considerations that don't take into account anything specific about the kind of object involved.
b) applying the categories is only supposed to generate knowledge when combined with stuff from the senses (namely: " the manifold given in sensory intuition")
Kant and Frege seem to have a notion of the actual = "wirklich" which only applies to objects you can causally interact with
Kant clearly accepts mathematical objects in some sense, but it's not clear whether he somehow thinks they are merely possible.

Idea: Many people find abstract objects spooky because they assume that they would have to be Wirklich, or something like it. The merely logical conception of object above doesn't require any such thing. So maybe mathematical objects exist in the logical sense i.e. we can state truths using singular terms for them and using quantifiers, but they are somehow not Wirklich.

ii. intuitability

Kant digression:
You use intuition to discover whether things could fall under it. [presumably round square would be an example of a putative concept that doesn't pass this test.]
geometric figures = forms of empirical objects
We can learn about them using intuition.

Perhaps it's an requirement that all objects are 'intuitable'?

defining intuitable
We will use intuition to mean a kind of perception that could apply to physical objects or abstract objects. We can distinguish
- having an intuition of an object, like perceiving an object (e.g. 'I intuit the equilateral triangle')
- having an intuition that some proposition about the object holds (e.g. 'I intuit that the interior angles of the equilateral triangle add up to 180')

Some issues:
-Should we require that one can have intuition *of* the object, rather than merely intuiting some suitable proposition about it? (call this strong intuitability) Or is it enough if you have an intuition of concrete objects that represent abstract objects, like the sequence of strokes Kant appeals to in his proof that 7+5=12? (call such a representation a quasi-concrete representation)
-On what sense does need to be possible to intuit something for that something to count as intuit*able*, and hence satisfy the requirement?

Idea cont. - Maybe mathematical objects are real in the logical sense, and intuitable, but not wirklich/causally effecations...

4. objecthood=having the logical role of an object

We will stick with Quine and Frege and say that the logical criterion (not wirklichkeit or intuitability) is all that's required for objecthood.

Some questions arise if you accept this definition of "object", about how to further spell out the view.

a) Which logic has the property that *its* singular terms and quantifiers correspond to objecthood? Maybe we should allow modal or other intentional notions, and if we do we will get different answers about what objects there are.
b) Maybe there are some entities which aren't objects? (i.e. maybe there's some important ontological category that's wider than objecthood - like some kind of meinogian being)
c) Maybe there are some objects which don't exist? (i.e. maybe there's some important ontological category that's narrower than objecthood - like fictional objects might be said to logically objects, but not really exist)

5-6 are about b and c respectively

7. Quasi-concrete objects

We will call abstract objects quasi concrete if they have a special relationship to certain concrete objects that 'represent' them e.g.
strings of letters --- inscriptions of strings of letters
sense qualities --- experiences of those sense qualities
shapes --- physical things that have that shape

We can look at the physical representatives, and keep in mind individuation criteria for the abstract objects. These individuation criteria say when two different concrete things 'represent' the same abstract one.

Some sets are quasi-concrete: sets with concrete ur-elements are represented by those ur-elements. But pure sets are not quasi concrete.

Overall Conclusion: mathematical objects exist in the logical sense, although they are not Wirklich, and although some of them are not intuitiable even in the weak sense allowed by looking at concrete objects that represent them.

Field on Normativity and Logic

In "What is the Normative Role of Logic" Field argues that you can't understand logic descriptively as (eg. the project of studying necessarily truth preserving syntactic manipulations), and so are forced to a more normative conception of logic (logic is the study of how one ought to reason), by the following dilemma.
-classical logics can't state a general truth predicate (if they could, we could inductively argue for the soundness of logic, and hence a consistency proof for logic L in logic L, contra Godel 2)
-non-classical logics which can state a general truth predicate, sometimes fail to preserve truth, in some degenerate cases (in places where good reasoning wouldn't lead you to in the first place).

So (Field says) the only people who can *state* the descriptive criterion for being a logic, deny that logic has to have that property.

But I think there's a gap in this argument: why should you have to be able to state your criterion for what a good logical system is, *in the formal language of that logic*? In particular, why can't the anti-normativitst about logic reply like this:

A. Classical Logic Version:

Logic is the study of formal systems of syntactic manipulation which are truth preserving for various fragments of our language (e.g. english sans any truth predicate, english sans any repeated application of the truth predicate). Practically speaking, this is all we need for almost every purpose except philosophy of logic and truth. And the moral of Tarski-Godel considerations above is that this is all we can get.

Formal, exceptionless, rules for truth-preserving reasoning are great when you can get them (i.e. for limited fragments of our language) but what Field has shown, is that we can't get any such rules that apply to the informal notion of truth (as opposed to the notion of truth-of-a-sentence-in-L, for various restricted L)

Admittedly, taking this route involves giving up the traditional and somewhat attractive Fregean idea that logical principles are fully general, and hence would apply to all possible reasoning, but - at least- this seems way less revisionary than the normative relativism about logic where Field winds up.

B. Non-Classical Logic Version:

It was indeed wrong to say that logic studies patterns of inference that are always truth preserving. Field is right that Logic studies patterns of reasoning that are truth preserving "where it counts". But "where it counts" doesn't mean something normative like 'with regard to premises that one could be justified in believing', but rather, something descriptive like 'with regard to premises that people are likely to every actually accept'.