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:
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.