Another Article

I seem to be on a kick of passing things along; here’s another article.

Dang, philosophers are smart.

drt.

Reflections Of, The Way Life Used to Be, Before 30.

Not so long ago, I turned 30 years old. Now, if you missed it, the truth is you probably didn’t know about it because I didn’t make a big deal of it. I let my birthday slide by because I don’t usually have good birthdays.

Usually, they’re associated with a long reflective analysis of how my life isn’t where I feel it should be. I tend to compare myself to other people, and usually that means that I compare my weaknesses to other’s strengths. At 30 many people have published seminal works, and I feel that I, too, should have. Of course, the truth is that I tend to compare myself to people I’m not really comparable to. I like to think (rightly or wrongly) that I’m capable of doing philosophy with the best of them–where ‘them’ refers to philosophers 30 years my senior, to people who’ve been teaching and examining the issues for more than twice as long as I have. Feelings of failure promoted by: prolonged singleness, the absence of a career, and any notable strengths, have not helped. I’m not world famous; I haven’t published a work that has fundamentally altered the direction of Western thought; I’ve not made millions with a great invention. And most painfully, it seems, never will.

I’m average, and I’ve spent a great deal of time trying not to be. Of course, I can pretend not to be–I can use big words and make references to obscure poets and thinkers–but to do so is to engage in one of the two things I hate about the academy. Ideas matter. I have pursed the education I have because of an interest in truth. Attempts to name drop, and quote obscure people don’t demonstrate intelligence; all they show is that you can read. Of course, there is a fine line–it is important to read great thinkers precisely because they have shaped where we are today, and articulate concepts found in popular culture with much greater clarity and cogency than the average lay person is able to. They are useful in exposing you to ideas, but shouldn’t be used as a sort of weapon to oppress those who haven’t read them.

I’ve always been far more interested in truth than I have been in being ‘right.’ I find arguing very difficult and try to avoid serious conflict when I can. Of course, though, I’m not immune to having my pride hurt, and acting in an over the top sort of way to defend it; even when I’m wrong. But, so are you–it’s called being a part of a fallen world.

I’ve placed myself here. I directly made the decisions that led to my current situation; my education didn’t happen to me, I chose it. What is happening in my life now is the outworking of previous decisions–some good, some bad–but all ones that can be traced to my free choices. I can’t, and don’t, blame anyone but myself; it would be cowardly, and destructive, to try to.

On past birthdays one often finds a depressing journal entry whining about the issues above. A good friend, perhaps my best friend, predicts that I’ll have a massive mid-life crisis, and suggested a 30th birthday meltdown was likely. But, for the most part a meltdown hasn’t happened. I do have worries about: the future, a career, my personality, and my character. But, they need not be primary; they need not be ever-present.

Life is good. Or, if we believe what theologians tell us, it is meant to be good.

Are You Married?

Today, I had a meeting with my guided study professor.

I’ve been reading a book on epistemology, and it talks about the roll of the Christian Scholar. I’ll develop the thoughts more in an upcoming blog–but one of the key points is that Christian scholarship should contribute to shalom, or ‘justice as shalom.’ [Shalom isn't possible without justice, but justice is not a sufficient condition for producing shalom.] I wasn’t exactly sure what to think of this suggestion. It challenged my current paradigm for academia, and is probably the most lasting thing I’ll take away from the book. I asked my advisor if this was what he though, and what his experience as an academic had been like. His reply began like this:

Him: “Are you married?”

Me: “No.”

Him: “Women tend to prevent men from getting too divorced from practical life in academics. It isn’t that these issues don’t matter, just that women tend to bring a much needed balance. You don’t see very many women mathematicians, or philosophers–it’s likely for this reason.”

[At this point he paused.]

Him: “Not everyone has to get married, though.”

Mathematics: An old Argument.

Originally, this was something I wrote in reply to a number of people on the internet.

I know, arguing on the internet is a foolish act. Nothing on it ever leads to intelligent or meaningful dialogue. Forums always degenerate into name calling and rubbish. But, at the same time, it keeps you sharp. Not sharp in the sense that you’re able to argue and win, but sharp in that it helps you to analyze other people’s thoughts and arguments–it engages you in a certain type of activity and way of thinking.

Below is what I wrote:

Simon et al.,

First, I’m going to re-hash some things because I want to be clear for everyone in this thread. In particular, I’m not doing it because I think any of you are slow–or dimwitted–or something. Clearly, I’m the one who is in the minority, and so clarity is of great importance. Furthermore, I have some questions that need answers. I’ve tagged them in this form (Q1), (Q2), etc. My questions don’t seem to get answered–so this should help clarify what I want answered and where it is to be found. Also, it would be helpful if any potential responders decide to challenge things head on. I’ve gone to the trouble of separating my reply into different sections labeled accordingly as “Reason 1″, “Reason 2″, etc. Please reply by clarifying exactly what “reason” you are replying to.

I am, as you’ve pointed out, committed to some form of mathematical realism and I do think numbers are real in some sense. I’m hesitant to say I’m a Platonist–lest I be misunderstood.

No one here has disagreed with me that mathematics is necessary for virtually all of science. I’ll take it that we all hold the common ground that:

(A) mathematics is indispensable to science. (Or, science cannot be done without mathematics.)

My statement that numbers are “invisible, not made of matter nor energy, timeless, mysterious, [possibly] contradictory, and everywhere and nowhere” was simply to point out that if I am right, then the category of things that are “invisible, not made of matter nor energy, timeless, mysterious, [possibly] contradictory, and everywhere and nowhere” makes sense–and has at least one member. If the category itself is valid, then one can’t reject God because of (or based on) these attributes (although, clearly other attributes might prove incompatible with these, or might on thier own be sufficient for rejecting God) Again–just so we’ve got this nailed down and can move past it: this isn’t an argument for God’s existence. The purpose here is not to show that somehow God is analogous to numbers, and therefore He exists–I am not trying to prove God’s existence at all. In fact, for the moment–lets just cease with the references to God. It doesn’t add anything here–as we don’t seem to be able to agree about numbers.

So, the existence of numbers in reality. I should just comment on what you’ve said here:

“I think I would have to agree with Jesse, Vadim, Magistra and the others that numbers are real in the same sense that the word “balance” is real. It is a human construct representing something real.”

I can agree with this in some vague sense–but that’s the problem. You’ve left it vague. “It is a human construct representing something real.” (Q1) What exactly does human construct mean? (Q2) And what does “real” mean in this context? I’m the one who is arguing that mathematical objects are real. Others have said they are human inventions, and are the product of convention. (Q3) I can put it bluntly: Is the number five a real thing? Not a thing that you can go pick up, but a thing none the less.

(**) First, Simon, you seem to have grasped that I’m not talking about the “words” or “labels” for numbers. Of course they are convention. Of course I think the label “one” could have been something else, like “uno”–or even something random like “genom.”

Reasons why I don’t find anyone in here who rejects my position convincing:

Reason 1: I don’t think numbers and mathematical statements, are analogous to words and language. (Here of course, I’m suspending the question of existence, with the hopes that we can reach an agreement over the relation between math and language)

For example, consider these parallel statements:

1) The tabby cat is brown.
2) The number two is prime.

Aside: Here we see that numbers are not “adjectives.” It is true that they can act like an adjective in some sentences–but 2, and similar sentences, show that it makes perfect sense to treat any particular number/set as a noun. Also notice, both sentences have the same structure. A noun (cat/number three) is followed by a verb (is) and then an adjective/property (brown/prime).

In 1, the truth value of the statement can be determined only in some broader context. Namely, if we know what cat in the world it refers to, and if that cat is in fact brown.

But in 2, the truth value of the statement is determined only by reference to the things or concepts in the statement. We don’t need to know what particular “instantiation” of 2 is being referred to, nor do we need to know any “external” facts. All we need to know, is that being prime, is a property of the number two. However, we should all note together that even though 2 doesn’t need to refer to the external world says nothing about whether or not numbers exist.

Now–maybe you’ll think I’ve drawn a bad example. (Q4) Do you have a better sentence, and what is it?

My best thought is that you’ll reply with something like this:

1′) The word snow has four letters.

Here, 1 does have a truth value that is determined only by the properties of the word snow. In this case, the label for snow is convention (We both agree to (**) above–so I assume you’d think the same for language). If we change the label used for “snow” (which we can do because it is just convention) to “broon”, then the truth value of the entire statement would be different. It switches from true to false. Here is one way it’s evident that mathematical statements like 2 are dis-analoguous to language. The truth of a mathematical claim is independent of the language used to convey it. For example, If we change the label for three to “uno” it has no bearing on the truth of 2. 2′s truth claim is expressed in language, but is determined by something outside of language.

Again–maybe you don’t like this example. Maybe the problem lies in the statement I’ve chosen as mathematical. Here is a different one:

2′) Nine is less than ten.

Here, it seems that you could conceivably argue that we couldn’t know what “nine” or “ten” are without appealing to the external world. Hence, I’m appealing to a broader context, just like 1. This argument is unconvincing. Of course I have to appeal to the physical world–we all agree that you can’t go find the number five lying around in an uninstantiated form. Just like you can’t go find gravity lying around either. That some things don’t have uninstantiated forms doesn’t mean they don’t exist. Anyways, the point is that if one understood the concepts of “nine” and “ten” then one could determine the truth value. But, that isn’t the case for 1.

In fact, lets dwell here for a minute. (Q5) How is gravity any different from a number? We all agree it exists. Explain how its existence is any different from a number.

Maybe the problem is that math is analytic, and so we need a corresponding analytic statement. We can try again:

1”’) A bachelor is an unmarried male.

Here, I certainly can concede that being an “unmarried male” follows from what it means to be a “bachelor.” And here, it is quite clear that the concept of bachelor is purely a human construction, and is a description. I see this as the only possible reply that you can make that is at all feasible to the thesis that mathematics is analogous to language. 1”’ leads me to my second criticism–we will return to this point though.

Reason 2: Something can only be a convention if it might have been otherwise, and can still be otherwise. Conventions are not unrevisable. Here, I’m drawing on (**) and (A). Words aren’t what is at the heart of the controversy. To say that mathematics is a convention, is to say that 2 (from above) might have been otherwise if we decided it to be so, and could still be so if we decided to change it.

For me to treat this position seriously, I’d need one of you to provide me with a good argument for thinking this is the case. It is not rational to say that we might have decided that modus ponens might have been different. (In fact, it is demonstrably false. If logic is a convention, then any attempt to provide reasons for thinking logic is a convention, will themselves rely upon logic. The thesis that logic is true by convention reduces to the claim that “logic is true by convention plus logic.”) To think that 2 (from above) is somehow revisable, is to fail to grasp how interconnected mathematics is, and how it is logically derived.

Furthermore, (A) which we seem to agree on, is itself incompatible with mathematics as a convention. Let’s suppose for a moment that I’m wrong, and you are right–that mathematics is a convention and that mathematical objects are not real. Then, it would follow that one of the most important tools science uses, relies on a convention which doesn’t describe anything real.

Now suppose you meet someone who is not sold on this whole “science” thing. She asks you why she should believe science if it relies on an arbitrary convention that could/can be changed. (Q6) How do you reply? Suppose she is a mathematician and knows that by Godel’s incompleteness theorem it is impossible to prove that mathematics is consistent. She points out that it is quite possible that our convention contains a contradiction–and that from a contradiction we can derive anything. She notes that if our convention had a contradiction, then we would be wrong about everything that we call science. She asks: (Q7) what grounds do we have for believing that math, as a convention, is consistent?

In science, truth seems transitive. If special relativity is true, special relativity can’t rely on some other theory that is false. Transitivity implies (through a backward chain) that mathematical theorems are “true”–but here is a problem. (Q8) What sense does it make to talk about the truth of something that is a convention? To me it doesn’t seem to make any sense. (Q9) Furthermore, what is to prevent us from using convention to establish anything we want–the answer can’t be reality, because mathematics is a fundamental part of how we interpret reality.

It might help to think of this in broad terms. We currently think of our universe in non-euclidian geometry. To say non-euclidean geometry is a convention, means that non-euclidian geometry could have been otherwise. Suppose for a moment that we collectively decided to change it–that would instantly mean that the theories we describe the physical world with would also have to change. Non-euclidian geometry is the framework for special relativity–we couldn’t just throw out part non-euclidian geometry, and keep special relativity as it is.

Another example would be this: we believe that there are an infinite number of primes, and that primes do not follow a predictable pattern. Modern cryptography borrows heavily on this fact. We could not, by convention, decide that primes do follow a pattern.

Now, what of 1”’? Well, clearly the word and meaning of bachelor are a matter of convention. Maybe this is analogous to 2? Well, no. It is entirely possible to imagine a world like our own in which there simply are no instances of bachelors, and where the concept would refer to nothing. A world populated only by women, fits the bill nicely. However, it makes no sense to imagine a world like our own, but missing the concept of the number two, or in which two is not prime, or in which the number two cannot be instantiated. Here, what is being hinted at is the necessity of mathematics. Language cannot be necessary because it is dependent upon humans.

As a result of the analysis of Reason 1, and Reason 2, I’ll consider the language option dead. I can’t think of any other ways to salvage this position. Perhaps one of you can breathe some life into it. I’m not sure how though.

Reason 3: In math, we don’t invent things, we discover/derive them. Here I’m describing what mathematicians do–this is a descriptive statement about what actually happens. Discovery implies an independent reality. If as you’ve been suggesting, mathematics is a convention, then what mathematicians actually do doesn’t make any sense. (Q10) How do you explain discovery in mathematics, if math is just a convention?

Reason 4: Tradition is on my side. Now, admittedly tradition is sometimes wrong. But, to have Plato, Kant, Frege, Godel, Quine, and Putnam all backing you is, well, comforting. I’m not sure how many people here know who all these people are. Of course, most of you will know Plato and Kant. I’ll use the most impressive example. Godel was accredited as being the greatest logician ever. A quote from Einstein indicates that he came to the Princeton Institute for Advanced Study so he could “have the privilege of walking home with Godel.” These aren’t mediocre intellectuals I’m claiming as support. You can’t dismiss me, and them, with some lame claim that we are confusing abstract concepts with reality.

Now, clearly my “Reason 4,” is an argument from authority–which admittedly doesn’t show the truth of what I’m claiming. But, we believe authorities the vast majority of the time. The burden of proof is placed on those who deviate from tradition, and usually a reason is needed to reject authority. (Q11) What is your reason for rejecting authority here?

So there it is. Four reasons I reject what has been said in this thread, and eleven questions that any good reply should answer. Again, My questions need answers. I’ve tagged them in this form (Q1), (Q2), etc. and my reasons are listed in this form “Reason 1″, “Reason 2″, etc. Please reply in a similar form–as I’m looking for clarity, and this seems the best way to get it.

You, reader, are unlikely to care. You might even think that much of what I’m talking about makes no difference, that it is abstract and devoid of any real importance. It doesn’t, after all, connect to everyday life. Let me give it meaning for you: if the category being debated is valid for mathematics, it is (at least initially at some basic level) coherent to talk about God. I don’t claim that this kind of God talk will change hearts; it may not even change minds. What it does do, though, is show that Christians are not cognitively deficient, or denying noetic duties. God talk is valid–even if we concede a great deal more to a skeptic who believes in reason alone than any right thinking theologian would.

If you remain unconvinced of the worth of this kind of thought, please await my upcoming post “The Christian Scholar”

d.r.t.