### The ‘Objectivity’ of Mathematics

#### by Witty Ludwig

The first thing I need to make clear is that I am not a mathematician. I’m not so sure it is an area in which I could ever have a great deal of proficiency. I have, however, become increasingly interested in the philosophy and psychology of mathematics. Given my deficiencies in this area, though, I welcome comments even more than in any other section because, being in such alien territory, I feel I am even more likely to err than in other cases.

It was during my early reading that I had a debate with a friend: I had argued that mathematics was not in fact objective and certainly no *more* objective than language, whatever that would mean. The context of this was a discussion of how, I believe, we have launched strange items into space in an attempt to communicate or demonstrate that intelligible life exists on earth, and one such item consisted of a string of mathematical formulas. I said this was misguided since it presupposed that another life would have mathematics, would have need of mathematics, which is entirely a human convention / cultural game. My friend didn’t necessarily disagree on this point but did proffer that I had to agree it was, at the very least, more objective than anything language could offer as a means of communication. And so our discussion began.

His point was, let’s say that you had three walnuts in front of you: no matter what word or sound uttered, the fact that, together, they comprise the concept ‘three’ is either very simple to explain or self-evident.

This part was relatively easy to dismantle. You only need to look so far as South America (let alone extraterrestrially), to Daniel Everett’s work with the *Piraha*, to see that it is a cultural tradition, habit, convention, etc. to need to group items into classifications, that is, and that cultures can exist that have no need and therefore no use for such a convention in their own life patterns. These people could not learn how to add cardinal numbers, despite asking to be taught, because they don’t use a calculus in their day to day lives: even their trading system worked on concepts such as ‘fewer’ and ‘more’, ‘larger’ and ‘smaller’, without the need for breaking quantities down into units.

— To retort that they are simpler, less civilised, less intelligent etc. would not be correct. Their faculties are the same as ours but their way of living is so far removed from how other areas of the world have developed, their way of living so different, that what is relevant to them is vastly different from what is relevant to us, thereby shaping their entire way of thinking, speaking, living. There is no reason to believe, or not to be able to conceive, that an advanced alien form of life could not function in a similar way– or an entirely different one from either scenario, for that matter.

Consider Daniel Tammet, for instance, who has what he describes as synaesthetic experiences as opposed to anything we would call ‘calculating’.

The second part, however, caused me to falter. He pointed to the rim of a circular drinking glass on the table and said: “Look, a mathematical formula can calculate the exact surface area of this glass. I can take the length from the centre to the edge of the rim, square it, multiply it by *Pi*, and I will discover the surface. This formula will work on any circle. It is axiomatic and will apply in any and every like case.”

I knew something wasn’t quite right with what he was saying, but I couldn’t pinpoint it. I wanted to say: You haven’t *discovered* anything. The circumference or, more accurately, that rim shape was already there to begin with. And this, I think, is the mistake being made. Nothing has been discovered because the proof is already present– the space defined by the rim, the length of the rim itself, were there from the start. All that was necessary was to create a system that maintained consistency. What more is needed? If, from the very beginning, the formula for a circle’s area was: *Pi*(r)squared – 1 , would this cause any problems for the engineers, architects, etc.?

Wittgenstein, *Remarks on the Foundations of Mathematics*, I.5., c.1937-1938:

“How should we get into conflict with truth, if our footrules were made of very soft rubber instead of wood and steel?– ‘Well, we shouldn’t know how to get the correct measurement of the table.’– You mean: we should not get, or could not be sure of getting, *that* measurement which we get with our rigid rulers. So if you had measured the table with the elastic rulers and said it measured five feet by our usual way of measuring, you would be wrong; but if you say that it measured five feet by your way of measuring, that is correct.– ‘But surely that isn’t measuring at all!’– It is similar to our measuring and capable, in certain circumstances, of fulfilling ‘practical purposes’. (A shop-keeper might use it to treat different customers differently.)”

What is important to recognise here is the importance culture and custom play even in something as seemingly inexorable as mathematics.

“… it can’t be said of the series of natural numbers– any more than of our language– that it is true, but: that it is usable and, above all, *it is used*.” (I.4.)

I am neither a mathematician nor a philosopher and I am not familiar with the terms of this debate, so this obviously puts me at a disadvantage here, but as regards the second argument, I thought I would attempt a friendly comment, but please let me know if I am misrepresenting the matter. I understood the argument to be able to be paraphrased as follows:

If certain mathematical formulae describe physical phenomena truly, then mathematics is an objective truth.

Of course, developing an axiomatic system which governs the physics of our universe is not a discovery, but a characterization of that universe. Do not the laws of mathematics determine a logical space, many possible statements in which, e.g. Pi(r)squared – 1, fail to characterize the physical laws instantiated in our actual world? Such formulae are nonsensical in relation to our physics, but that is a fact about our world, not about mathematics. That is, a mere subset of the set of all mathematical statements characterize the laws of physics. However, does this fact undermine the “objectivity” of mathematics?

Certainly it does if mathematics is taken to be objective just in case all meaningful statements of mathematics are meaningful physical statements. But it does not if mathematics is taken to be objective just in case all its statements are (synthetic) true a priori, or just in case some other condition is met. Not really being initiated into this debate, I cannot offer a more plausible such condition, but for the sake of discussion, take that suggestion as a rough approximation of one.

A possibly relevant thought (that I’ve read in a secondary source and whose origin I don’t know): Some early proponents of the validity of deduction from first principles including Descartes took the logical consequences of these principles to determine numerous possible ways the world could be, only a small portion of which are actually ways the world actually is. What distinguishes logic and pure mathematics from physics and the other empirical sciences is that it is the task of the former to deduce this logical space, while it is the task of the latter to determine which of those possibilities are instantiated in our world.

Thank you for taking the time to comment; I really appreciate the input. I just wanted to clarify a few of your thoughts first before launching into a response.

In terms of the context: I suppose the debate was more fundamental; i.e., that mathematical rules are axiomatic, objective, true. Addition, for instance. I appreciate, though, that the particular facet of the argument that I have described could be paraphrased as such.

Is your point here to separate mathematical propositions from empirical ones? E.g., taking Euclid’s parallel postulate or, indeed, its converse, for example, the mathematical proposition itself could be seen as objectively true despite the fact that we couldn’t, empirically, test its truth due to being unable to construct infinite lines that demonstrate its ‘truth’? Is this what you are trying to distinguish?

On this last point, since you pair mathematics and logic together (I’m not suggesting unwisely), is it safe to assume you feel that mathematics is subject to logical inference? That is, that 2 follows 1, 3 follows 2, 4 follows 3,

ad infinitum, is a logical and necessary process?Note: This is the same user as the one who offered the original response. As that was my first comment on WordPress, I failed to realize that it would be posted under the name of an old, humorous blog, which never actually got used.

It was indeed my intention to distinguish between mathematical and empirical propositions. Let me demonstrate the thought process which led me to this, and I will expose why it fails.

I distinguished between the two because they are subject to proof of a fundamentally different kind: the former to proof by deduction, including mathematical induction, and the latter by less rigorous inductive reasoning. It is my understanding that the method of mathematical induction does not merely suggest the truth of the conclusion sought, but rather constitutes rigorous proof of it. In contrast, inductive reasoning in the empirical sciences merely suggests the truth of the conclusion, and it is open to refutation by counterexample. In scientific practice, moreover, it is frequently the case that such a counterexample can be provided. To my mind, the notion of the form of proof is a valid criterion on which to base a distinction between mathematics and logic.

This brings me to your second inquiry: Do I feel that mathematics is subject to logical inference.

I would hedge this on the success of specific axiomatized regimentations of mathematical systems in a logic. As regards, for instance, the natural numbers, I understand the endeavor of providing a proof of their consistency to be complicated. Famously, Goedel’s incompleteness proofs require that one look outside Peano arithmetic for methods to prove the statements made in his theory of numbers. It seems to me that the success of applying rules of logical inference to mathematics is only as good as the logical system regimenting it.

Sorry, it was very late when I posted that last comment, and now that I read it over again, I see that it isn’t very clear as written. A few clarifications:

1. Paragraph 2: I do not feel that an argument to distinguish mathematics and empirical science fails. Rather, the complete union of mathematics and logic is not something I know how to defend, or if such a program is, in fact, tenable. Independent of this, however, mathematics occupies a privileged position relative to, say, physics, as a mathematical system is adequate in any possible world, while a physical one might not even be adequate in our own.

2. Paragraph 3, last sentence: change “between mathematics and logic” to “between mathematics and empirical sciences”.

3. I do not see proof as the sole sensible criterion to distinguish between these, of course. Other criteria could be advanced as well, such as any which distinguish formal from empirical sciences.

Sorry for any confusion. I’ll try to be clearer in the future.

Not a problem at all. I admit, I did struggle to follow your preceding post but appreciate the time you put into typing it.

Just to clear any misrepresentations on my part, I didn’t mean to imply that mathematics wasn’t ‘objective’ due to my friend’s second argument failing; only that I felt his point was misconceived in thinking that did make it ‘objective’.

Was your point, when intitially commenting, that you feel mathematical propositions, as opposed to empirical ones, are objectively true? I ask because I didn’t intend to imply that there isn’t distinction between the two types of statement, in case you felt that was my point. I feel that your second paragraph, in your preceding post, that is, is spot on.

I should add, though, before I trap myself and get too caught up: I don’t think, linguistically, it makes sense to say whether mathematics is ‘true’ / ‘objective’ or not. I think that ends up causing a muddle in and of itself.

I see now that I misunderstood what you originally asserted in refuting your friend’s argument. I thought you meant that the failure of the second argument constituted direct evidence for mathematical unobjectivity. I see now that you meant it simply is an instance of a consistent system, and thus merely fails to constitute positive evidence for mathematical objectivity.

Because I thought that you considered the failure of his argument to implicate the unobjectivity of mathematics, I offered that first comment to the effect that empirical and mathematical propositions are to be distinguished. I wanted to assert that mathematical principles are in some sense prior to the implementation of those principles in particular systems such as euclidean geometry, and hence, the (un)objectivity of mathematics is not to be determined by an examination of any one particular system. This is a position I would commit myself to, for the purpose of debate at least.

As regards language, I do agree that a theory of language has little to say about the objectivity of mathematics. It is taken for granted, for instance, that prior to 1995, a speaker of English educated in mathematics could only felicitously say, “It is mathematically possible that there exist no positive integers a, b and c s.t. for any integer value greater than or equal to 2, an + bn = cn.” (superscript n’s) After Fermat’s Last Theorem was proven, however, it became felicitous to begin instead with “It is mathematically necessary that …”, this in spite of the fact that it was always the case that this proposition was mathematically necessary. I believe this to be fairly evident.

But our mental representation of mathematical truth may be relevant to linguistic semantics in certain other ways. Let me attempt to explain, for instance, a somewhat technical counterexample to a popular theory of focus, where focus is understood to be a phenomenon is natural language by which the truth conditions of a sentence are altered by intonation. In the following example, capital letters informally indicate phonetic stress.

(a) Nine only is the square of THREE.

(b) (Imagine Abby introduced Bill and Cathy to David. No other introductions were made.)

“Abby only introduced BILL to David.”

By standard assumption, “only” has truth-conditional effect here, quantifying over a domain of alternatives. The relevant alternatives are given below for (a) and (b).

(a) is the square of negative three, … , etc.

(b) introduced Cathy to David, … , etc.

The technical problem can be demonstrated thus:

1. The constituents that “only” quantifies over are not just “THREE” and “BILL”, but rather predicates “is the square of THREE” and “introduced BILL to David”. (This is motivated by distinct data.)

2. Viewed extensionally, “is the square of three” and “is the square of neg. three” are the same in all possible worlds.

3. An extensional theory of only falsely designates (a) and (b) as true.

3. Enter intensional semantics (relativizing to possible worlds). Though intensionalizing the truth-conditional meaning of “only” will correctly make (b) false, the mistake is actually only corrected when the meaning of a predicate varies between possible worlds, and Manfred Krifka, who has written on this topic, maintains mathematical ones do not.

Out of curiosity, would you agree that this notion of mathematical truth is linguistically relevant? Sorry if I didn’t explain that well enough or if I sounded pedantic at any point. I assume you aren’t familiar with the focus literature.

No need for any form of apology; I appreciate the discussion.

Although I’m certainly not familiar with the authors whose ideas you’re drawing from directly, I think understand you well enough.

When you say linguistically relevant, do you mean relevant to linguistics? In what sense? For ascertaining truth conditions? If so, for me it is not. The example you gave, to me, seems very Tractarian and, I would bet a handsome sum of money, would certainly have followed on from the analytic tradition in the style of Frege, Moore, Russell, Whitehead, etc. and, whilst its development and its merits are still a hot topic, I think its misguided and opt with Wittgenstein’s post-Tractarian thinking and influence.

I appreciate that doesn’t give you any specific material to contemplate or engage with but we will end up typing to each other at length forever! Do feel free to email me, though, or let me know if you would like me to email you.

I’m sure I’ll explore my thoughts in this area in another post soon, which will no doubt result in further discussions!

I figure, from reading bits and pieces of L.W.’s treatment of mathematics, that the meaning of ‘five’, say, is the use we make of the word in empirical propositions such as ‘I ate five apples’. So mathematical objects are really a product of our linguistic evolution (numbers exist only because we have a need for grouping similar things).

However, once those objects (or, concepts) have come into everyday use, aren’t the rules that govern their use such as 2+3=5 objective in some sense? Isn’t it necessary that any alien civilization that uses numbers have the same addition rules as ours?

If so, inscribing mathematical formulae on the Voyager tablet does make sense. If aliens use numbers and can make the connection between their and our symbols, they may recognize the “objectively true” formulae. Don’t you think?

Thanks for taking the time to comment. I agree with your first paragraph; the second one I’m not sure that I do and so I just want to pause there.

I suppose it depends what you mean by ‘objective in some sense’? From how you phrase it, language would surely be ‘objective in some sense’ too, if your criterion is just that rules govern every day use?

“Isn’t it necessary that any alien civilization that uses numbers have the same addition rules as ours?”

Well, again, trying to explore this idea: I’m not quite sure what you mean by ‘using numbers’. Do you mean the symbols themselves or the processes of calculating, subtracting, multiplying, etc.? I think for us to say that they are doing any of these things, if we’re to interpret ‘using numbers’ as the processes I mention, whether they uses different symbols or similar ones, our way of life would have to be *similar enough* that we would recognise their activities. I.e., we would note that they too have a system for money, for exchange, and that this involves quantities; or perhaps we would see how they gesture to objects, hear what they say aloud, and what they do with the objects subsequently (like your apple quotation above).

Once I reach your third paragraph, I’m not sure we standing on mutual ground by this point because, as you can see from the way I’m approaching this, the formulae won’t necessarily have meaning. For one, it presupposes that they live in such a way that they use numbers at all and, secondly, even if they did, that their rules of application would be similar(1). As for your last sentence, wouldn’t a Rosetta Stone of some sort be needed? Or are you assuming our symbols are identical to theirs?

(1) I think L.W. gives as an example somewhere that, for instance, if you were to take four objects (pretend these ‘1’s are simple straight lines):

1 1 1 1

and try to explain ‘2+2’ to someone by grouping the lines into separate groups of 2 ( 1 1 ), someone unfamiliar with our system of calculation might suggest that these four objects could add up to the number 6:

(1 + (1) + (1) + 1)

2 + 2 + 2

Where the first two lines are added, the second is added to the third, and the third is added to the fourth.

Which equals 6.

(Hard for me to do on the internet!)

We would then correct them by saying ‘Ah, I’ve seen what you’ve done there but that’s not how we use addition; we do this…’

I’m sorry if this footnote doesn’t make much sense. It’s much easier in person with a pen and paper.

Thanks a lot for the detailed response. Thanks to your illustration, I now see how the form of life influences even such ‘objective’ notions as rules of addition.

I’m ploughing through the Investigations currently, but I find L.W.’s mathematical philosophy especially intriguing. Hopefully, ‘a light shines on me from above’ and I get around to his Remarks on the Foundations of Mathematics.

I find your writing similar in style to Wittgenstein’s. Perhaps an influence of the great man?

I suspect so– when most of these posts were written, I would have been reading his works furiously. Not that I would encourage it but, for the last few years, my reading has been very narrowly confined to his writings and his associates; inevitably, I’m sure a certain level of mimicry is going on whether I realise it or not, embarrassingly.

I’ll paraphrase but I remember one of his comments somewhere, in a despairing tone, that “the only legacy [he] is likely to leave is a certain type of jargon”, which was aimed at students I think trying to speak like him. I’m always worried of falling into that camp, if I haven’t already.