Psychology is no nearer related to philosophy, than is any other natural science. — Shawn

I think I understand what Wittgenstein is trying to say here, but I think I disagree. As I understand it, the purpose of philosophy is to help us become more aware of how our minds work and how we think. What could be more relevant to psychology than that?

The theory of knowledge is the philosophy of psychology. — Shawn

Isn't this consistent with what I wrote just above? Doesn't it contradict Wittgenstein's previous statement?

I would like to point at a real life example of possibly what Wittgenstein would have agreed with. — Shawn

I don't see how your discussion of knowledge and education and their effect on identify is an example of what Wittgenstein discusses.

My personal belief is that knowledge is a form of "memory" encoded in the brain, more specifically the hippocampus. — Shawn

If you are saying that knowledge is actually stored in the hippocampus, I believe that is not correct, although it is true that the hippocampus has an important role in memory.

With the process of education a person carries the memories of what they ought to do or become in a form of narrative that educators present about how the world works or latter in one's formative process what domain of knowledge a person is apt at in relation to the narrative of the educator. — Shawn

As I understand it, although education may play an important role in forming our sense of identity, it is not the primary mechanism. A lot of our sense of self comes from our human nature built in by biology and neurology. In addition, much of who we are comes from what we learn from our day to day experience of our lives, especially when we are children. Most of that learning takes place with no particular intention or narrative. People with no formal education, unless you count normal socialization as education, still develop a sense of self.

I find education as one part of the puzzle of identity theory, or at least the part of the puzzle which is quite possibly the most important part of the bigger picture, — Shawn

Again, I don't believe this is true.

Apart from education as a formative process in the young mind, I would like to ask the reader about how does the reader suppose that knowledge can influence one's identity? My personal belief is that knowledge is a form of "memory" encoded in the brain, more specifically the hippocampus. With the process of education a person carries the memories of what they ought to do or become in a form of narrative that educators present about how the world works or latter in one's formative process what domain of knowledge a person is apt at in relation to the narrative of the educator. — Shawn

None of that really resonates with me. I have no significant sense of what I ought to do, only what I want to do or what might get me into trouble if I do it. When we actively engage with education, it often seems to be about providing ad hoc justifications for matters which already appeal to our preferences or to follow our disposition. We enjoy math - we study math. We enjoy reading - we study literature. Or in my case, don't enjoy anything at school - so I skive off and go into the city each day to smoke and look at buildings. But if you are just talking about school, say, for me it's the experiences and relationships that help shape identity, not the formal learning. My identity is based more on intuitions, attractions, repulsions and pleasure seeking than any conscious sense of knowledge.

My personal belief is that knowledge is a form of "memory" encoded in the brain, more specifically the hippocampus. — Shawn

Of course knowledge is stored in memory. There are only two forms of memory: short- and long-term. The short term memory is information. If it's used right away - like a postal code or sale price - it's discarded immediately after; you cannot recall what cantaloupes cost in August, 2004. But if your Grade 4 teacher was any good, you remember the 9X table. That's knowledge in long-term memory.

But that knowledge consists of all kinds of things. The name of your pets, of doughnuts, constellations, dead movie stars, cars, sporting events, chemical elements, philosophers, restaurants; how to park uphill, what pie charts mean, what a jerk your boss is, how hot sand feels on your soles, that you shouldn't wear white after Labour Day, how much to tip a cab driver, the taste of mashed potatoes, what never to say to your significant other unless you want a fight.... Knowledge is all the useful and precious and unwelcome clutter in your memory. There is no index or handy catalogue: you have only a vague idea what-all is in there; half of it is only available when you're deeply asleep.
Of course every scrap of knowledge contributes to your identity. You grow with each experience, with every fact you learn, with every skill you acquire.
The down-side is, when you forget things, your identity diminishes.

what does the reader think about the quote from Wittgenstein and the role of education and learning on the development of the person or individual in terms of their psychology and "identity"? — Shawn

Education is just another part of life. If it's formal, you learn conformity, discipline, compartmentalization of subject matter, some social skills, a respect for or resentment of authority, depending on your school(s). You also learn many things that may be useful through your whole life and many others that you need only until the exams are done. You won't always know which is which until thirty years later when you discover you can correct the rival who misquotes Hamlets' soliloquy or you need to make a tent out of a canvas sheet. Aside from the influence of the school environment on your attitudes, education is just more stuff deposited in the memory banks. If it's informal, education is simply instruction and experience. Whatever environment it takes place in will influence your attitudes.

All experience adds to identity.

In my view Wittgenstein's comment of "Psychology is no nearer related to philosophy, than is any other natural science" just refutes the idea that philosophy would be psychology. At Wittgenstein's time psychology was quite new, actually.

Today psychology is understood as an interdisciplinary field of study and close to sociology, not purely a "natural science".

Apart from education as a formative process in the young mind, I would like to ask the reader about how does the reader suppose that knowledge can influence one's identity? My personal belief is that knowledge is a form of "memory" encoded in the brain, more specifically the hippocampus. — Shawn

Just like there's a difference between data and information, I think there's also a difference between "memory" or memorizing something and the knowledge (and the understanding) of something. And this understanding defines how we act, how we view the World and things in it, which surely then passes on to the distinguishing character or personality that we have.

I think there's also a difference between "memory" or memorizing something and the knowledge (and the understanding) of something. — ssu

Albert Einstein: “If you can't explain it to a six year old, you don't understand it yourself.”

If you can't illustrate a concept or process by applying them to specific examples, you probably don't really understand them.

Richard Feynman: “I don't know what's the matter with people: they don't learn by understanding, they learn by some other way — by rote or something. Their knowledge is so fragile!”

Memorizing without understanding is like sex without orgasms.

I would like to ask the reader about how does the reader suppose that knowledge can influence one's identity? — Shawn

Years ago I completed a curriculum and became a meteorologist and practiced that discipline. Later I completed a curriculum and became a mathematician. I drifted away from the former and lost that identity, while becoming identified with the latter. Meanwhile I became proficient at rock climbing, and that was a separate identity. I married and had a child, so I became a parent. We may have multiple identities.

Psychology is no nearer related to philosophy, than is any other natural science.
The theory of knowledge is the philosophy of psychology.
Does not my study of sign-language correspond to the study of thought processes which philosophers held to be so essential to the philosophy of logic? Only they got entangled for the most part in unessential psychological investigations, and there is an analogous danger for my method. — Shawn

BTW - what is he actually saying here, it seems vague.

I didn't find Wittgenstein's quote helpful.

Perhaps review some psychology? There are 4+ kinds of memory: working memory, sensory memory, short-term memory, and long-term memory. The hippocampus converts short-term memories into long-term memories by organizing, storing, and retrieving them. It also helps you remember emotionally significant events by connecting with the amygdala, which is involved in emotional responses. [And that's a critical piece of how we learn what we ought to do.] The hippocampus helps you with spatial spatial navigation, and learn about your environment and be aware of what's around you. The hippocampus helps you remember what words to say (like, is that my amaryllis or my clueless mistress?) And it plays a role in emotional processing, including anxiety and avoidance behaviors.

One of the problems of identity formation is that it is (in my opinion) a global process beginning with biology and encompasses all manner of informal experiences over many years. Education -- especially as a formal project organized by societies -- probably plays a small role, except, perhaps, where the educational regime is highly intrusive and rigorously directed.

In terms of my identity, I was born in the upper midwest as a male homosexual. I learned to become gay. I was born into a specific social / economic milieu, and absorbed what I would later call a working class identity. I thought I wanted to be an English teacher, but found I was altogether unsuited for that job. Still, being an English major is a piece of my identity. Even though I have been rooted (stuck?) in the midwest for 76 years, a piece of my identity involves 2 years of experiences on the east coast. .

Because education is a mass process for most people in the industrialized world, it can't make that great a contribution to our identities. Further, people who have never gone to school (can't read, write or do sums) still have an identity.

To what extend do you feel you have been aware of, or preoccupied by your identity/ties over time? Is identity just a given that you don't really consciously explore, or is it something which you often think about? Personally I don't really have a strong sense of self unless I end up stuck at a function or dinner party and am made aware of how little I share with others - in terms of interests and inclinations.

I have spent a lot of time thinking about identity. Class, sexuality, religion, politics, education, etc. all figure into this. It seems like I have become more of an outlier over the decades. There are, of course, other outliers with whom I share commonalities, but I am no longer circulating enough to come into contact with many of them. I don't even know where I would begin to find them, face to face, these days.

I'm content, though. At 78, identity has receded as a concern. It is settled, and that's good.

The one area where identity might still be an issue is age. I liked being young and able much more than I like being old and unable. I used to feel physically robust; now, not so much. Intellectually and emotionally I feel at the top of my game, such as it is. Self-appraisals are notoriously inaccurate, of course.

Personally I don't really have a strong sense of self unless I end up stuck at a function or dinner party and am made aware of how little I share with others - in terms of interests and inclinations. — Tom Storm

Personal identity is, of course, personal but it is significantly shaped by all the different kinds of interactions we have with others. Contrasting identities can sharpen our own identities--such as those in a social gathering where one's dinner companions seem more like aliens from the vicinity of Betelgeuse than Australia or the US. Or sometimes we fit together like obviously matching puzzle pieces.

Thank you.

The one area where identity might still be an issue is age. I liked being young and able much more than I like being old and unable. — BC

Yes, that's an interesting one. The kids at work now seek me out as a relic of a pervious era - 'What were the 1980's like, Tom? They must have been really cool.' I've never felt young, only inexperienced. Now I realise ignorance is forever and don't mind so much.

Interesting posts. I assumed that most interlocutors would agree with the notion that during our most impressionable years, such as 3-5 years old, when preschool starts, and then kindergarten, and then even elementary school, and then high school, and then (for some) college, and then graduate school, and then post-graduate school, well I don't want to labor the point that we spend quite a lot of time in the educational system.

Yet, I see that most users agree or disagree that knowledge and education seem to have an impact on their identities.

However, I suspect that some people take their psychologies as a form of identity. If true, then what would it mean to profess one's psychology as an identity? Or maybe the right question would be, to what degree is an identity related to one's psychology?

Thanks for the posts.

It's a very broad question. At root and primarily what you learn through the education system can be contextualised by viewing that system as a microcosm for society as a whole. And what you learn in that context is not the knowledge that is transmitted (and knowledge transmission even within the domain in which it is transmitted is no longer considered a satisfactorily comprehensive description of the function of that domain), but the way in which power is distributed through the distribution of knowledge and how this creates structures through which social life is controlled and regulated.

Another way of saying this is it's not the knowledge that is primarily identify-forming in an educational context, it's the context itself as a way of framing knowledge as power that forms the soclal identity and the ground on which individuals' navigation of this embedded framework rests. You might call individual strategies for negotiating the framework individual identities. But the framework is what's primarily internalised and grounds them.

the framework is what's primarily internalised and grounds them. — Baden

:cheer: :fire:

when preschool starts, and then kindergarten, and then even elementary school, and then high school, and then (for some) college, and then graduate school, and then post-graduate school, well... — Shawn

A few education commentators have asked "just why are a significant number of people spending at least the first third of their lives in school?" And that's not about medical doctors who go through medical school, internship, residency, and more before they become the narrow specialist of their dreams. Scholars of all sorts can spend a very long time in school.

"Student" as a way of being in the world?

Regulating the labor pool?

Long term milling of human raw material to achieve low friction parts?

It just takes half a life time to learn enough?

Produce effective servants of the technocracy?

I don't know. Pre-school and primary school might have a greater impact on individual character than anything that happens after the pupils are 14 years of age on up. Certainly, schooling has effects, often benefits, for students, as long as they are pursuing something in which they are interested and have a stake. That isn't always the case.

For an unknown percentage of students, education is a treadmill. In some areas, students hop off (or are thrown off) well before graduation from high school. A significant portion of college enrollees do not complete bachelor degrees. Graduate programs--whether they are high quality or just credential mills--maybe have a balance between input demands and pay offs. Not sure. But a lot of potential PhDs seem to wash out just from the long dreariness and uncertain job prospects after completion,

A 14 year old ambitious, talented, and well-funded student probably already has the essential characteristics to succeed in the long run, and eventual become a law partner, sought after architect, engineering whiz, the famous doctor, and so on.

Identity formation isn't the same for everyone over a lifetime. Those who turn out to be very successful at education and subsequent careers probably began high school with the essential striver-identity in place. The identities of the people who don't attend college (about half the population) are going to be formed differently than the strivers'. Those who miss the boat (in ever so many ways) are going to have another dissimilar experience of identity formation.

I was never a striver. I hadn't planned on college. I rarely had very clear occupational objectives at any given point. For me, 4 years of college was a turn-around experience -- more so than it was for many students. College made an unusually big difference in my self-appraisal, but my experience wasn't typical.

BTW, you have hatched yet another very worthwhile topic.

Another way of saying this is it's not the knowledge that is primarily identify-forming in an educational context, it's the context itself as a way of framing knowledge as power that forms the social identity and the ground on which individuals' navigation of this embedded framework rests. You might call individual strategies for negotiating the framework individual identities. But the framework is what's primarily internalised and grounds them. — Baden

I want to "yes, and" this. Primary and secondary education is also an unholy union of parenthood and peer socialising. It is the "village which raises a child", if that village were administered entirely by beleaguered and hopelessly overworked academics, forced to play the roles of parent and prison warden. The social machine that reproduces knowledge is also the principal site that society collectively reproduces itself within.

The social machine that reproduces knowledge is also the principal site that society collectively reproduces itself within. — fdrake

Astute and well said. So... just as a snide comment, is this secondary education meant to seem like "work"? I mean, the kind of work that we do to make and make due with bills, etc.?

Not sure. But a lot of potential PhDs seem to wash out just from the long dreariness and uncertain job prospects after completion, — BC

Forty to sixty percent of American PhD candidates do not complete their degrees. Many of these encounter a roadblock in the fundamental aspect of the degree: Doing original research. But there are various other reasons. The percentage is a bit higher in the humanities. Lower in STEM disciplines.

Thanks, Jgill. If the damn rats don't cooperate, what's a biochemist to do?

Not in STEM, but certainly the humanities have an oversupply of candidates for job openings. Colleges can be choosy about filling adjunct positions on up to tenure track jobs. Enrollments are down in quite a few of the majors. I think studying literature and languages is eminently worthwhile -- intellectually -- but not so much for employment. In the good old post-WWII days, one could study literature for its joys and still get a secure job teaching.

Visual Capitalist chart of ROI on various degrees:

That chart surprised me for two reasons:
 
1) Why does psychology have negative return on investment? I thought psychology was very important in the USA and since the American government doesn't provide medical attendance with taxation but with insurance, it surprised me that it is in negative. Nursing and health professions return on investment is $195K. Why is psychology not located in the latter?
 
2) Why are legal studies separated from public administration and social services? I would put them all inside the bubble of social sciences. Law is considered a degree of social science and public administration here.

I think Wittgenstein is quite right.

Psychologists (those who stay in their lane, anyway - and a lot of them don't) study human behaviour.

Philosophers don't.

For example, psychological egoism is a psychological thesis (a very implausible one). It states that all humans are motivated purely by self-interest.

Ethical egoism, by contrast, is a philosophical thesis (also very implausible). It states that we ought to be motivated by self-interest.

They have nothing to do with one another.

First, glad you were not washed away in the great flood. Floods cause such great loss. "Come hell (heat) or high water" as the saying goes.

There are several reasons why psychology has a negative return. One is that not all psychology majors work as psychologists. One might work in a psych hospital as an attendant, but one doesn't necessarily need a degree for that, and the pay is not great. A second reason is that a bachelors degree in psychology does not enable one to "practice" as a licensed psychologist. For that you need at least a graduate degree in social work or some kind of doctorate. Insurance companies require state certification as a licensed therapist before they will pay you. Probably a lot of psychologists end up doing other kinds of work -- like graduates in a number of majors do.

The term "legal studies" isn't very clear to me. When I think of 'lawyers' I think of 'law school' and 'law degrees'. There are people who train to do 'legal work' for lawyers, but who are not members of the profession as such. Don't know.

Yes and, to develop the idea, there’s a kind of bait and switch whereby what’s ostensibly offered, “knowledge” (of this and that) is offered as an implicit route to power, but functions to obscure the actual route to power (the meta knowledge of the system of power in which such “knowledge” is misleadingly elevated) both in terms of its content and mode, i.e. this “knowledge” tends towards a static “body of knowledge” that divides the individual against itself rather than an integrated praxis that would unify and dynamize it. And this exclusion of praxis, the inculcation, not just of disembodied “knowledge” but of the idea that knowledge (implicitly generalized as power) just is disembodied “knowledge” enables the gradual castration into the social that the social needs to inflict to reproduce its organs (institutions and those willing to be dispensable cells therein).

The process runs from the moment we are told to sit down in our groups and listen to the teacher to the moment we receive our high school diplomas. So, by the time we get to university and read Foucault and Nietszche or whoever, it’s too late. They too are castrated down to just more “knowledge” because we can’t undo the psychological sedimentation the education system inflicts on us with more sophisticated versions of the same mode of sediment even if, in abstract terms, it blows everything we thought we knew open. There has to be something else, from somewhere else, to crystallize meaningful opposition, and that’s rare.

So, the social reproduces itself through the education system by a process of immunizing itself against its own elementary structures—us—so that we may be subsumed in the organs that make it work. And it does so by training us against learning in any meaningful sense that would foster individual power and foment effective dissent. Instead, what we’re offered is a life of confusion where learning becomes either learning to integrate further into a diseased system that doesn’t want us qua individuals or learning for leisure, a form of relaxation / game of pretence that allows us to imagine we are doing other than we are doing (being integrated, subsumed, castrated), the former a direct route to individual annihilation and the latter only a distraction from it.

Techno-consumerism fits very well with the above as it feeds on this passive knowledge enculturation, transforming it into an almost ubiquitous opportunity to commodify us for advertisers, creating social cells split by our need to escape the suffocating banality of such without the means to actually do it. If there’s any “good” news, it’s that the more effective a society reproduces itself and exploits its own “human resources”---us— the more effective it (eventually) tends to be in exploiting other societies for its benefit—them. So, we get to be passive consumer worker bees, but at least fattened by our own honey and that of others.

First, glad you were not washed away in the great flood. Floods cause such great loss. "Come hell (heat) or high water" as the saying goes. — BC

Thanks for your concern, friend. Appreciate it. Everything seems to be up again normally, but the number of deaths is still shocking for most of us. :broken:

Well, coming back to the topic...

The term "legal studies" isn't very clear to me. When I think of 'lawyers' I think of 'law school' and 'law degrees'. There are people who train to do 'legal work' for lawyers, but who are not members of the profession as such. Don't know. — BC

Exactly. That's what I thought too. To become a lawyer, it is obvious that someone needs a law degree. But the law degree itself is useful and goes beyond being just a lawyer. Firstly, judges and prosecutors should be included in 'legal studies' as well, but there is also a bubble (chart) saying public administration. Where should judges and prosecutors be located then?
On the other hand, I wonder what the requirement is to represent the state of the U.S.A. legally regarding legal international affairs or conflicts. We have a special bureau called 'Abogados del Estado' (State of Spain Lawyers). They are considered part of the public administration and not lawyers per se. I remember that Spain and the USA had a trial on the civil rights of a boat*. Spain was represented by Abogados del Estado on American soil. But who represented American sovereignty? I guess not a random lawyer but a legal public expert from the Bureau of Justice.

*Plot twist: We won the trial. It was a big surprise here because we didn't expect to win in the U.S. Supreme Court.

Even though I find education as one part of the puzzle of identity theory, or at least the part of the puzzle which is quite possibly the most important part of the bigger picture, what does the reader think about the quote from Wittgenstein and the role of education and learning on the development of the person or individual in terms of their psychology and "identity"? — Shawn

I have pulled out a 1942 math book to share with a child I am tutoring in math. Why a 1942 book? because the focus is on practical math. Many number stories are used to present math so a child can relate to what is being learned. That is opposed to math which is so abstract there is no human relationship with what is being taught.

Also, a 1942 math book because older books are full of moral lessons. Again and again, children read about being considerate, thinking of others, having good manners, etc.. It is a terrible error to believe that in the past schools taught only reading, writing, and arithmetic because it all came with lessons about a highly moral culture taught with no need to mention religion. In 1958 we replaced that education with education for technology and adopted the German education model of education that led to Hitler. Education for technology left moral training to the Church as Germany did and today we think morals are a religious thing, not a logical thing. We have forgotten when morals have to do with liberty. We are in a real mess!

Nothing is more important than education. Education is like a genii in a bottle. The defined purpose is the wish and the students are the genii. The result of replacing the education we had with education for the Military-Industrial Complex is having a popular national leader who talks about half the nation being the enemy within and promotes hatred and violence. People are now focused on working for money more than a sense of meaning and purpose and intrinsic values. We are not judging each other by character, but by how much wealth and power they have. In the past, we thought virtues were synonymous with strength. Today we want guns.

That is a pretty significant change in how we identify ourselves.

Yes and, to develop the idea, there’s a kind of bait and switch whereby what’s ostensibly offered, “knowledge” (of this and that) is offered as an implicit route to power, but functions to obscure the actual route to power (the meta knowledge of the system of power in which such “knowledge” is misleadingly elevated) both in terms of its content and mode, i.e. this “knowledge” tends towards a static “body of knowledge” that divides the individual against itself rather than an integrated praxis that would unify and dynamize it. And this exclusion of praxis, the inculcation, not just of disembodied “knowledge” but of the idea that knowledge (implicitly generalized as power) just is disembodied “knowledge” enables the gradual castration into the social that the social needs to inflict to reproduce its organs (institutions and those willing to be dispensable cells therein). — Baden

Absolutely!

You paint a picture of knowledge as a monolithic yet untethered abstraction.

Monolithic, in the sense that the static body of knowledge pervades society totally. It saturates us as both assumed common sense, and that sense's unfulfilled expectation. It determines those who know what we know. And those who do not. The enculturation of this knowledge must also remain relatively static in order to reproduce the social forms that that knowledge pervades and engenders. For such expectations, in order to function as an assumed body of knowledge, need a means of propagating the assumption as well as its content.

Also untethered, a subject which satisfies the demand of that knowledge, one who knows what we know, satisfies it in a normative sense rather than a semantical one. One does not need to be able to calculations with fractions to count as someone who knows how to do calculations with fractions - most jobs need a qualification that says you can do this, most people never learned to do fraction maths. There is a divergence between what is known and what is counted as known by whom. Whatever tethering mechanisms distribute satisfaction of these knowledges into bodies thus only reproduce this knowledge approximately, but also therefore are not required to reproduce this knowledge in toto. Thus we're left in a situation in that who counts as knowing what we know is distributed by the broader social form of education, rather than the content which it is designed to reproduce. The medium of education is to a large part also its message.

Every person is not, however, a coordinator of the medium of education. But they are a vehicle for its expectations - and thus its norms. Everyone who knows what we know expects others to know what we know, for that is what it means to know what we know. Those expectations however are stratified, as the broader social form of education distributes who counts as knowing differently from who knows. Whenever someone thus passes on what we know, they pass on the stratified means by which those expectations of knowledge are formed. Alienating us from their own capacity to reproduce a social form of knowledge, as its guide.

I believe you can feel this alienation when teaching students. There are daily moral dilemmas, some are related to the curriculum and some are not. Regarding those which are not related to the curricula - you know your students are doing things that are socially non-normative but are nevertheless morally permissible. You know they are thinking and acting in ways that are socially unacceptable in broader society but work in their community. You're thus confronted with the responsibility of raising concerns regarding deviations from what is normative and socially acceptable, regardless of its moral status. Because you know your students will be punished for deviations, and thus act as an organon of that punishment - reproducing the expectation by enacting it. There are even forms to fill in when something non-normative is disclosed. Not that they always are.

Regarding knowledge of the curriculum - the kind of student that satisfies all benchmarks in a subject has already been discouraged from pursuing their insights and skills due to herding them through the curriculum at a required pace. It is not uncommon to see an allegedly failing student have a profound insight, which you nevertheless cannot spend time developing with them since it is more important to their life to count as knowing what we know than teaching them how to learn, to grow their own capacities and insights.

Regarding knowledge of the curriculum - the kind of student that satisfies all benchmarks in a subject has already been discouraged from pursuing their insights and skills due to herding them through the curriculum at a required pace. It is not uncommon to see an allegedly failing student have a profound insight, which you nevertheless cannot spend time developing with them — fdrake

I agree somewhat with the "required pace", but you and I know that in post-K12 math and related subjects it takes effort and time to accumulate a background necessary to advance or apply knowledge even a bit. In my years of teaching college math I have encountered only one such individual - an older student who dropped out to support himself as a poker player. He had taken my course in complex variables, and I recall speaking with him informally in the math office in which he brought up a really interesting and unusual notion on the subject, spur of the moment. Like a light bulb burning bright. I was unable to convince him to continue the curriculum.

I agree somewhat with the "required pace", but you and I know that in post-K12 math and related subjects it takes effort and time to accumulate a background necessary to advance or apply knowledge even a bit. In my years of teaching college math I have encountered only one such individual - an older student who dropped out to support himself as a poker player. He had taken my course in complex variables, and I recall speaking with him informally in the math office in which he brought up a really interesting and unusual notion on the subject, spur of the moment. Like a light bulb burning bright. I was unable to convince him to continue the curriculum. — jgill

I think our bar for profundity is different. Kids trip up on stuff like:

a ) x+1 = 2, solve
b ) x+1 = 3, solve

because they think that x in the context ( a ) is the same as x in the context of ( b ). Which is a very good reason to be stuck.

They also get tripped up on stuff like addition in the integers being commutative, but subtraction in the naturals not being.

Sometimes they even do things like 1-2 = 0, which isn't "wrong", per se, it's a sense of subtraction that removes instances from collections of stuff - like scoring off bags of pasta from a shopping list. You then have to tell them that it's "wrong", in some sense, to say 1-2=0 even when such subtraction is perfectly cromulent.

What about division by 0? This stuff got me so confused in calculus. Also, this 0.(9)=1 is also confusing because people don't view it as an ever expanding series, which is a contextual issue between a non-converging series as a limit (AS DEFINED by stating 0.(9)) and a converging series as a limit towards infinity. What matters in my opinion, is the law of identity in clarifying how these mental cramps arise.