My ponderings on reductionism have led me to think that reductionism is mostly correct, and partially a confusion.

Reductionists think that there is an objective reality, which is a mathematical structure. We have subjective impressions (qualia) which give a sense of this objective reality. Consider, for instance, a table. I can see the table from various angles; I can touch it; and so forth. Each of these subjective impressions gives me partial information about the table. The table itself, on the other hand, is something that I will never experience. The appearance of the table is not the table; the table is not an experience.

The table is a mathematical structure. But if I think of the table’s mathematical structure, that thought once again is not the table. A thought of a mathematical structure is not the mathematical structure itself.

To solidify this conclusion, think of the number two. You may have spoken the word “two” in your mind; you may have imagined two dots next to each other; you may have visualized the printed character “2;” you may have thought something more abstract. All of these are thoughts of two; but none of them are two.

What is two itself? This depends on whether or not we think that two objectively exists. In the first case, where two objectively exists, two is beyond all appearances of two, in the same way that the table is beyond all appearances of the table. Two is not an experience. We can, so to speak, see two from different angles (the word, the two dots, etc.), but we cannot see the whole of two.

In the second case, if two does not objectively exist, then it is only a formality of our language, a concept with no tidy correspondence to reality. It is a confusion to ask what two is, in the same way that it is a confusion to ask what a hipster is, or what rudeness is.

Under reductionism, some set of mathematical structures has an objective existence. These structures are something that we will never experience, just as in the first case we never experience two.

So under reductionism, there is an objective reality of mathematical structure, and we have subjective experiences which are themselves unreal. Through these experiences we can know everything that there is to know about the objective reality; but we can never experience the objective reality itself.

That said, we can imagine a mind which was able to think a thought that contained all possible knowledge of reality. We cannot ourselves see a table from every angle simultaneously; but it is possible to imagine a mind which can see three-dimensional objects in an instant, in the same way that we can see two-dimensional objects in an instant. Similarly, we cannot ourselves think of the whole mathematical structure of a table in an instant; it is too large and complex. But we can imagine a mind which can do this.

So we can imagine a mind which can think everything at once, and perceive the entire structure of reality in an instant. Let us refer to this instantaneous perception of the whole of reality as the “final appearance.” Our metaphysics will give us a sense of what the final appearance would look like. For instance, under reductionism, the final appearance would be a thought of a vast and complicated mathematical structure.

Our concept of the final appearance is an inference based on all of the knowledge we have. We arrive at our concept of a table by looking at it, touching it, thinking about it, etc. By synthesizing these different experiences we conceptualize the “final appearance of the table,” which is our idea of what the table is. And we construct our idea of the final appearance of reality in the same way.

But reductionists throw out some of the information in constructing their final appearance. To see how, empty your mind of all thoughts and look at your hand. Here we have an appearance of a hand, which is just as valid as the appearance which is a thought of the hand’s mathematical structure.

Why should the final appearance of the hand be an appearance of a mathematical structure, and omit the purely qualitative visual impression of the hand? Why do we want to say that the mathematical structure has reality, while the visual impression does not?

Why do we want to say that the objective reality is a mathematical structure? We gain our concept of objective reality by aggregating appearances; but our appearances are not only appearances of mathematical structure, but also entirely different appearances. Why do we throw out the non-mathematical appearances in constructing our picture of reality, in constructing our final appearance?

If reductionism means simply that mathematical structure is everywhere, then reductionism is perfectly correct. But if reductionism means that mathematical structure is the only real thing, then I see no reason to believe that it is correct.

  1. #1 by Eric on March 7, 2012 - 8:02 pm

    A believe a mathmatical structure could describe those visual qualities you speak of. The visuals could be considered the rendered or compiled result of the mathmatics, similar to how a computer can render a visual scene. But your train of thought has merit in that I am uncertain that mathematics could be used to articulate feelings, which we know and feel exist. Maybe the reductionist tosses out feelings for this reason, or perhaps uses mathematics to explain the chemicals that are present in human brains when such feelings are reported?

  2. #2 by antitheology on March 8, 2012 - 12:32 am

    You’re right that math can describe a lot of visual qualities. The main part of vision that specifically feels to me like it’s irreducible to math is color. Sure, we can formulate various theories of color, but ultimately the subjective “redness” of red feels to me like it’s not captured when I write #FF0000 or whatever else. When I make that switch, haven’t I stopped talking about red and started talking about #FF0000?

    I disagree with the idea that the world is made of mathematics, but it seems to me like an idea that’s not *obviously* wrong. I think the idea has a lot of weight to it, actually, and I’ve struggled with it extensively. At this point in the battle in my mind, non-reductionism has won, but it’s a precarious victory. The way forward would probably involve teasing apart the issues in a more subtle and detailed way than what I’ve done so far. If you’ve actually resolved an issue then there shouldn’t be any strained feeling when you think about it; and there shouldn’t be any feeling that you’ve thrown anything out or biased the issue in any way.

  3. #3 by Eric on March 8, 2012 - 4:21 pm

    I think maybe what we could say is that in the first place, stuff exists. Whatever else we can say about it is meaningless until we assign some sort of context in relation to other stuff, and until we formulate some way to describe it to ourselves and others (language, math, illustrations, sound vibrations, hand gestures or whatever else). I’m not sure if that’s reductionism or not, but I always think of reality as this gooey and hard to define “something” that we as humans have mentally divided, categorized, named (etc.) for our own understanding. But that understanding can never encompass and define the whole, as you pointed out in the table example. The mind of a new-born infant is in many ways less distorted where perception of reality is concerned, simply because it absorbs all information without bias (or comprehension, perhaps!)

    So I think what we run up against when we think of the color red is that there are many ways we can describe it. HTML code, vibration frequency, in comparison with other colors… But because our human perception is limited by languages we sometimes come up short with ways to express everything we see and feel, such as the “redness” feeling you describe.

    What do you think?

  4. #4 by antitheology on March 9, 2012 - 1:43 am

    I understand the idea you’re getting at with the “stuff” of reality. (Philosophers try to dress this intuition up with a lot of fancy words like “substance” or “phenomenon” but clearly we’re just talking about stuff.)

    I agree that we do need a language for describing reality, and mathematics is a language which can say a hell of a lot of what we’re interested in saying. You could perhaps sum up the reductionist thesis by saying “mathematics is a language which can describe anything.”

    Of course there’s an important distinction to make between the description and the thing described. A cat is not the letters c-a-t. There is no isomorphism between the furry feline and the sequence of characters. But when we describe something using mathematics, we can construct a description which is isomorphic to the thing itself. If we model a system of particles, our model contains the very same information as the system itself. We haven’t just given a label to the system; we’ve used our language to go inside the thing and reveal its structure, all the details of what it is.

    So we have a distinction between just giving something a label, and making a linguistic construct which actually explains the thing from the inside out. Now when we talk about the qualia of red, that subjective sensation of the redness of red, I can’t see any mathematical way to go beyond merely giving that a label. I can’t see how to reach inside it and reveal its structure as a mathematical formula. Sure, maybe a mature science of the mind could do this — but could it? Because it seems to me that ultimately any mathematical formula can’t really be isomorphic to red without in some sense *being* red; and mathematical formulas are never red.

    #FF0000 is an inside-out explanation of phenomena like frequencies of light coming out of a monitor. But I think as far as being a description of a qualia goes, it is a mere label. (The same is true of the word “red” of course; in my opinion we have nothing but mere labels for this phenomenon in our language.)

    Anyway, that’s roughly my line of thinking on this issue. I can’t see how to construct an isomorphism between redness and a mathematical object; but I could very well be confused somewhere, as that’s how philosophy works.

  1. Reductionism II « antitheology

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