Three levels of mathematical reasoning

Well, I said I was going to take a one-week break from the internet, but I’m drinking now and that implies blogging, which is a contradiction. QED

There are three levels of mathematical reasoning. The first is purely concerned with the mechanical reproduction of simple algorithms, which is concerned with practicing rules in order to get a result. This is something like computing the following sum:

6354664161
+ 54754678

This sort of thinking develops early. I will call this first level “fundamental” mathematical reasoning because I am firmly of the opinion that drilling fundamentals is the key to mastering any endeavor. Neuroscienmagistically, I believe it is primarily concentrated in a module in the left hemisphere of the parietal lobe. However, the speed of acquisition of new algorithms is governed by g (the ability to produce new white matter), and the speed at which algorithms are applied (IQ according to timed tests) is due strictly to mental energy expenditure and existing white matter.

Fundamental mathematical reasoning breaks down and produces frustration when it is unclear how to apply an algorithm. Taking exponents for example, most primary school students are taught that the expression 5^6 = 5*5*5*5*5*5. If you tell them to evaluate 5^(-1/2) they will become anxious.

The second level is concerned with something called the “concept image” in education research. It concerns perception of forms, both visual and abstract, for the purpose of deciding which algorithms and heuristics to apply to a problem. This is the level at which most calculus students operate, and it is the level at which undergraduate physics is taught because students are assumed to lack a philosophical background.

(For the record, I believe it is a mistake to teach science in the absence of philosophy. Or statistics, for that matter. Would you believe that most hard science curricula do not include statistics courses? It boggles the mind.)

Repeated practice of an algorithm produces an awareness of how objects behave under an operation. For instance, when solving for x in the following equation…

3x^2 – 5 = 0

…we eventually cease to think or speak of “adding five to both sides”. Instead, we imagine a mental movie where the 5 symbol jumps over the equality symbol and takes the place of the zero. Without recourse or memory of specific “rules”, we simply absorb “allowed behaviors” of symbols as if they are physical objects. Throw a rock and it falls to the ground, throw a -5 and it lands as a +5 on the other side of the equation.

I will call this “conceptual” mathematical reasoning. I believe talent in this area is a function primarily of visuospatial ability and discernment (rate of intuition, or rate of form -> heuristic). Hence, it is determined primarily by existing white matter times gray matter, and rate of gray matter production (which implies anxiety via amygdala stimulation and the reflective period to down-regulate anxiety, as previously described).

This sort of reasoning breaks down when the concept image (the absorbed heuristic) does not match the concept definition. A great example is the question of whether the function 1/x is continuous. Well, let’s look at the graph. Is it continuous?

The answer, surprisingly to most of us, is yes!

Calculus students tend to understand “continuity” as being equivalent to “I can draw it without lifting my pen” (pencils are for poor people). It certainly looks like there is a discontinuity at x = 0. But this is not the real, actual definition of continuity. The definition is that the right and left-hand limits of f(x) are equal for all x in the domain of f(x). And x = 0 is not in the domain. So it doesn’t matter that the limit of f(x) isn’t groovy at 0, because 0 isn’t in the domain. I can imagine some of you out there don’t still don’t believe me. Look it up.

The third level, known as “advanced mathematical reasoning” to decadent hipster bourgeoisie, is concerned with these concept definitions. I will call it “abstract” because that’s what it actually is. This is proper philosophical thinking, where a person checks their answer against the axioms, definitions, et cetera that they have graciously accepted as true beforehand.

The amount of gin remaining in my bottle suggests I don’t have much time, so I’d better wrap this up.

Advanced mathematical reasoning is equivalent to philosophical reasoning, except the colloquial understanding of philosophy has it to be a soft science, full of squishy and feelz. This is simply not true. Philosophy is only and always about local coherence: that is, definitions, and keeping large systems of definitions self-consistent. Every definition is a local generalization of some phenomenon, after all. Blimey, this is hitting me all at once.

Advanced mathematical reasoning means that you always go back and check your answer against the proper verbal definition, rather than relying on visualizations and heuristics. If the definition doesn’t fit the visualization, you have to change one or the other. Some local contradictions are allowed in mass association heuristic intuitionistics land, where King Global Coherence is maximized, may he live forever, but not in advanced mathematical reasoning metaphysics land. Mere conceptual reasoning is for level 2 thinkers.

Abstract reasoning seems to be mostly a frontal lobe thing, but this is usually specialized for social stuff. And crikey, is that a lot of stuff. Systems of behaviourisms and whatnot. Probably this is why genius requires an endogenous personality But it is also a parieto-frontal integration thing, hence the heavy correspondence of genius to IQ (~70%, according to Expert Analysis by Yours Truly).

There are forms of genius which correspond to transitions from each of these sorts of reasoning:

Fundamental -> Conceptual: Hey guys, I noticed something after counting up my corn stalks a million times. You know how long that takes! Well it turns out you can just do this thing I call “multiplication”.

Fundamental -> Abstract (uncommon): Hey guys, I noticed that when I add a row and a column to a square crop, it is also a square crop with opposite parity. See, one row and one column is one, which is “odd”, and two rows and two columns is four, which is “even”, and three rows and three columns is nine, which is “odd”, and so on. Let’s call this weird thing “induction”.

Two real examples of this: My humor is sadism post, which occurred to me as a series of English words in my mind that I had to make sense of afterward, and 2) Hamilton’s revelation of the quaternion formula.

Conceptual -> Abstract: Hey guys, let’s call all these similar-looking things by the same name. Let’s say “goat”. They also have different names like Larry and Bill and Persephone, but they also have this name “goat”. I know, two names for the same thing is pretty weird! But hear me out for a minute…

Pretty much everything that counts as “concept porn” falls into this category. Gervais principle, MBTI, that sort of thing.

Abstract -> Conceptual (statistics): Gosh guys, all these things with the same shared second name “goat” tend to eat our lunches if we leave them next to the “goat” pen. Let’s not do that anymore.

Abstract -> Fundamental (laws): Everybody has to punch the guy who lets his goats out of their pen. They keep eating all the corn! If we all hit him I bet he’ll stop doing that.

Lardy I hope that made sense. PUBLISH BUTTON.

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About Aeoli Pera

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14 Responses to Three levels of mathematical reasoning

  1. Heaviside says:

    I wrote this big long thing on mathematics before I realized that I misunderstood you, so I will simply say that philosophy is not about keeping large sets of definitions consistent. If it was, then then why are there so many aporetic dialogues? I have to deliberately resist the urge to use mathematical metaphors for philosophical ideas because I know that there is a fundamental gap between the two that leads all such metaphors astray.

    • Aeoli Pera says:

      You should post the big long thing anyway :-D.

      I will rephrase: the analysis of philosophical ideas is largely concerned with keeping systems of definitions logically consistent. Is that closer?

      • Heaviside says:

        “I think you are completely wrong here. What is mathematics? What is mathematics, historically understood?

        I think we should take “mathematics” to mean the mathematics of Faustian civilization, as Spengler described it. It is a historically limited and time-bound project. What is the object of this mathematical study? We can say that it is shapes and quantities, but when we say that we implicitly assume a mathematical definition of the objects of mathematical study. The real question, the one that reveals the essence of our mathematics, is what is the object of mathematical study considered outwith mathematics, that is, physically, historically, and philosophically? This task is very counter-intuitive, because, for instance, in order to answer just the first question we would have to be able to consider physical objects without reference to all of the mathematically-founded notions of modern physics.

        Well, I don’t have much more other than hunches about the answers to those questions, but if I open a book like Analysis, Manifolds and Physics the aim of things like set theory and measure theory is to build up and support differential geometry and topology. Now you might object to that book as obviously being partial to certain subjects just based on the title alone, and obviously you can find plenty of books by specialists that are about nothing but set theory and combinatorics and category theory and so on, but what is the point of all these subjects? These specialists all believe that what they do is worthwhile in itself, but I think the ultimate source and hidden spring of these subjects is the quantification and manipulation of res extensa. For example, abstract group theory that doesn’t have to do with the study of real condensed matter or space-time is just an” extra appendage that has been brought along for the ride.”

        That is what I wrote down, but I think the second paragraph isn’t very good. I will try to put things more simply, because I don’t have a good grasp of this issue, being a dilettante.

        Is there a limit to mathematics? My best guess is that there is something, some historically limited and time-bound project which Spengler identified as “Faustian mathematics,” but just putting a label on it does not tell us what it is. I heard once that a student’s understanding of a theorem is never greater than his understanding of worked examples. Vladimir Arnold said that mathematics is the branch of physics where the experiments are cheap. I think that mathematics is “empirical” in the sense that is a study which has an object outside of itself, though that object is not necessarily “physical” and the methods used to study it are not experimental. We can say that this is a “Platonist” position, but again, putting a label on it does not tell us what it is, and it is definitely not simply “Platonism” because we have to contend with the historicity of the object, because we have to identify something more specific and particular than universals like numbers.

        Hegel dismisses mathematical proof as totally unsuited to philosophy(which I think he is right to do) because mathematics deals with the purely external characteristics and relationships of an object, for example, the quantity or location of an object has no essential connection to what that object is. This would lead us to say that mathematics has no particular object, because quantities and distances and shapes can apply to any kind of substance. However, I have the suspicion that if we were to consider only a particular mathematics of a particular culture instead of mathematics in all times and places then we can say that that particular mathematics was the study of an object or a restricted class of objects and not all possible objects as we would have to concede to a universal mathematics.

        If “Faustian mathematics” is the study of an object, then it is not just a bunch of pure abstractions, in which case mathematics is not just a process of further abstracting and generalizing, but is primarily a process of intuition and discovery to which formalization and abstraction only plays a supporting role.

        My gut tells me that “Faustian mathematics” will come to completion in a century or two, and what replaces it will be as different from our physics as our physics was from that of the schoolmen and the alchemists and Aristotle. The question which occurs to me, is “is there any limit to the technological malleability of the material world?” I think that “Faustian mathematics” is not just mathematics but also our modern science and technology, and that these disciplines will reach some sort of completion with it, and they will closely approach this limit, which may be infinite. I think that technology will reach the point where there remain certain things we cannot do, but that what stops us is not natural laws, but something metaphysical, and then the whole metaphysical paradigm within which our technology exists will be aufheben. When this thing comes to completion, we will have a complete technical description of what the object I have been inquiring into is, but we can still think about it now, before that happens.

        >I will rephrase: the analysis of philosophical ideas is largely concerned with keeping systems of definitions logically consistent. Is that closer?

        No. Philosophy is thaumaturgy.

      • Heaviside says:

        Don’t tell me that you are going to prompt me to write all that and not give me any feedback.

      • Aeoli Pera says:

        Okay, I’m beginning to understand this now. Essentially, you are saying that Faustian mathematics may turn out to be a closed system, and if so we just haven’t figured this out yet. And the scope is already well-defined by axioms which are actually intuitive rather than logical, because we are implicitly concerned with investigating real objects whose properties we haven’t really defined except by playing around with them and saying “you can do this but not that”.

        Therefore, if we ever succeed in getting all the big parts connected, we will be unable to proceed except by starting over with new intuitive objects that we allow to behave entirely differently.

        My personal definition of math, and therefore the One True Definition, is the science of solving puzzles.

      • Aeoli Pera says:

        Maybe it would be more correct to say the science of unraveling puzzles.

      • Heaviside says:

        >Essentially, you are saying that Faustian mathematics may turn out to be a closed system, and if so we just haven’t figured this out yet. And the scope is already well-defined by axioms which are actually intuitive rather than logical, because we are implicitly concerned with investigating real objects whose properties we haven’t really defined except by playing around with them and saying “you can do this but not that”.

        >Therefore, if we ever succeed in getting all the big parts connected, we will be unable to proceed except by starting over with new intuitive objects that we allow to behave entirely differently.

        Basically.

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