Pure Mathematics is not Science   8 comments

This post was inspired by a recent discussion about “mathematics vs cryptography” over on Bruce’s blog, which I found particularly interesting given the discussions going on regarding the nature of scientific theories in my current graduate school class.

There is a fundamental difference between an axiomatic system (like, say, Geometry) and a Science. In an axiomatic system, you have a number of set definitions and a number of statements that are *given to be true* (these are the axioms). The practitioner then formulates a theorem, and proves the theorem to be true using the tool of logic, on the foundation of the axioms. If one of the logical consequences of the theorem contradicts one of my axioms, then the theorem is not true.

Compare this to Science, where instead you have a theory, which you attempt to prove or disprove by running experiments, gathering data, and analyzing the data in the context of what is considered to be the body of knowledge in your particular field. Science is the iterative process of trying to explain observable data using propositions that enable you to expand your capabilities to predict the outcomes of future events.

Mathematics is a *constructive* process – you are attempting to build a logically consistent system. In a real sense, it doesn’t matter at all if your system has any current practical use (the example I gave on Bruce’s blog is that there are no currently known applications for N-space topology). Of course, people will probably be more interested in your theory if you can show it has some real world application, but if your system is consistent, that’s all that is really required.

Science is a *de-constructive* process -> you are attempting to *derive* a logically consistent system given a bunch of experimental data and a bunch of additional theories that are supported in turn by experimental data.

This is one of the fundamental disconnects between a large percentage of non-scientists and people who practice science for a living; people who don’t understand how science actually works think that science is axiomatic; something is either true, or it is not true. Science isn’t like that, kids. As my high school AP Physics instructor said in class one day, “If you’re looking for Truth, go take a Philosophy class”.

Misunderstanding this is where the oft-repeated statement, “[Some particular scientific theory] is JUST A THEORY, you can’t PROVE that it’s true!” comes from. In an axiomatic sense, this is correct, but in a scientific sense it is totally irrelevant, because Science is not an axiomatic system. I can’t PROVE that the Theory of Gravity is true in an axiomatic sense. I can, however, prove that the theory of gravity is useful and therefore ought to be accepted as part of our our default understanding of the universe (well, to be precise, I have no desire to do this, but you can read plenty of Newtonian mechanics and physics and if you don’t get it, you’re just not a scientist).

Someday, someone will come along that has a better Theory of Gravity (Einstein did this – ed note), one which explains some things that the original Theory of Gravity did not explain. If those are useful things (they are), and there is no evidence that contradicts the newer Theory of Gravity, then scientists will adapt the newer, improved Theory of Gravity. Yay! This is how Science advances.

[Edited to add 08-22-2008] – another version of this post over at Cosmic Variance, with a follow up here.  Both are good reading.

Advertisements

Posted September 28, 2007 by padraic2112 in noise

8 responses to “Pure Mathematics is not Science

Subscribe to comments with RSS.

  1. There are no scientific truths?

    But more seriously: you point to the axiomatic nature of mathematical theories as what sets them apart from scientific theories. I think there are several problems with this. Most problematic of all is the fact that most mathematical theories (notably, elementary arithmetic) cannot be given a complete axiomatization. There is also the unreasonable empirical effectiveness of mathematics to be contended with. I’m just not sure that I’m happy putting math on the non-science side of the academic divide, then again I guess there is some question about why this should matter.

  2. > There are no scientific truths?

    Absolutely, or no, depending on how you define “truth”.

    If you mean, “Are scientific models internally consistent”, then yes, I believe that this is (or rather, can be) true. If, however, you’re asking, “Are scientific models objectively true?” then the answer is no, of course not… at least, not outside the bounds of their described behavior.

    One of the reasons that science is so horribly misunderstood by non-scientists is “have you heard that {some new theory} proves that {some old theory} is actually not true?”

    Relativity does not *disprove* Newtonian mechanics, it simply shows that the model of Newtonian mechanics breaks down when you approach boundary conditions (very large mass, very high speeds, very small distances). This doesn’t mean that the model is useless, merely incomplete.

    > Most problematic of all is the fact that most mathematical theories
    > (notably, elementary arithmetic) cannot be given a complete axiomatization

    Not really relevant. One of the consequences of Gödel’s work is that people (ed note: by “people” here, I mean “mathematicians” in particular) have realized that closure is not a required principle of a mathematical system; not all theorems must be provable with your axioms for the system to still be consistent. Consistency is, in fact, far more important than closure.

    > I guess there is some question about why this should matter.

    The main reason why I put mathematics in a non-scientific set of academic disciplines (I think that there’s more than one “divide” – “humanities” vs “sciences” is too baldfaced a comparison) is that in math, we declare our axioms. In science, we *don’t know what they are, we’re trying to discover them*. Fundamentally, this leads to different problems and appropriate solution spaces.

    But you’re right, this is largely the realm of philosophy 🙂

  3. You may be right when it comes to theory change and/or the question of comparing or updating scientific theories. Perhaps it is a mistake to describe this change or updating as getting closer to the truth. The shift from Newtonian to Relativistic physics might have been more like an extension or addendum to the old model. I can buy that. But it still seems to me that it is true that light moves at 299,792,458 m/s. Do you disagree that this is true?

    Interesting closing comment. This seems largely right and I think it is definitely in the realm of philosophy (and is nonetheless interesting for all that).

  4. > But it still seems to me that it is true that light moves at 299,792,458 m/s.
    > Do you disagree that this is true?

    Well, not to quibble semantics, but this is less of a “truth” than a simple fact – and again, it’s contextual.

    We know that in experiments we can devise, we have a huge body of evidence that tells us that this fact is universally correct (at least, in all vacuums). But, of course, that’s within the body of evidence we have, generated by experiments we can devise. For all intents and purposes, we *call* this fact as a generalized universal truth.

    But it’s not really a “truth” – there’s more to the speed of light than its speed in a vacuum. This number is not universally correct, light slows down in other mediums. So if you just generally assume that this is always the speed of light, you’re also generally assuming that it is always traveling through a vacuum.

    There’s a subtle philosophical and logical difference between saying that “this fact is true” and “this concept is a truth”. For the most part, I think the disjoin between non-scientists and scientists comes from the non-scientists thinking that facts are concepts, and the scientists thinking that facts are truths.

    The general populace, if they’re troubled to think of it at all, thinks of the term “truth” in the sense of the Platonic ideal. Scientists use the concept of truth to describe how facts and models explain the world, but this dual use of the same general language causes things get muddled.

  5. I guess I still don’t entirely get it. I’m coming from a philosophical tradition which is anything but Platonic about truth. As I see things, facts are neither true nor false, they are not the kind of things which *can* be true or false, the facts are just the facts. It is our statements or our beliefs which can be true or false. So if a statement expresses that P and it is a fact that P, then I would think that statement is true.

    You make a helpful point about context. Admittedly the unqualified formulation of the speed of light is false because there are cases of movement of light that are slower. Maybe if we include all possible contextual details we arrive a simple unqualified truth, e.g. the speed of light in a vacuum is 299,792,458 m/s. Now is that statement true? (Part of the reason I ask is because of my genuine and admitted ignorance about scientific matters, a sad state of affairs for an academic, I realize). It just seems that once we have nailed down the facts we have in our hands a body of statements which are true. The difficult part is in the nailing down of the facts.

    It strikes me that perhaps the reason you are hesitant to go along with this way of talking about things is precisely because of some Platonic overtones to the concept of “truth”. If we all agree to drop Platonism in favor of a kind of pragmatic and deflationary use of the concept of “truth” is it then okay to talk about what’s true?

  6. > It strikes me that perhaps the reason you are hesitant to go along with this
    > way of talking about things is precisely because of some Platonic overtones
    > to the concept of “truth”.

    That is a major factor, yes.

    > If we all agree to drop Platonism in favor of a kind of pragmatic and deflationary
    > use of the concept of “truth” is it then okay to talk about what’s true?

    Sure.

    The problem is, who is “we”? You and I can agree on this, but for the most part this is simply not how most people are comfortable defining the concept of “truth”.

    One of my gripes about the uses of the words “objective”, “subjective”, and “truth” is that they aren’t well understood (even inside of philosophy, where they belong), and as a result non-scientists start to regard science as constantly in a state of “incorrectness” -> e.g., “Sure, they say that evolution is true, but they said that nobody could break the sound barrier or split the atom, too, and those were supposed to be true.”

    Scientists need to be able to use clear unambiguous language to describe science, and none of those words are unambiguous.

  7. Pingback: “Nothing’s Impossible” « Pat’s Daily Grind

  8. Pingback: The Iron Law of Confirmation Bias « Pat’s Daily Grind

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

%d bloggers like this: