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Context:

There was recently a question on Math.SE: Inferior to Other Younger and Brighter Kids which starts as follows:

I'm a high school student (Junior/Grade 11) and I'm currently studying Spivak's Calculus on Manifolds. I've finished Single/Multi Variable Calculus (the basic ones without analysis/rigor), the first half of Rudin for Real Analysis and Linear Algebra (Lang) through self-study.

And then the OP writes how inferior he feels compared to other younger and brighter kids. Perhaps there are some deeper issues involved, but I don't want to dwell on it, nor discuss the situation of OP publicly.

The issue:

The issue I would like to write about is that more and more often I notice kids studying advanced books. Surely gifted students will access more advanced material earlier than their peers and I would applaud the kids reading almost anything they want. Yet, I have the feeling that sometimes this is just too early and reading advanced books without certain level of mathematical maturity may cause more harm than good.

My experience isn't enough to draw any conclusions, but this is not just a feeling, I've witnessed negative effects of such behavior multiple times. Here are some example motives:

  • Over-reliance on the already-learned material. There was an introductory problem given which could be solved using some elementary method. The gifted student thought "this is boring" and solved it faster using an advanced method. Then there was another problem given, but the advanced method couldn't be generalized to this new problem (while the elementary way could) and the gifted student was unable to find a solution until the end of the class.
  • Missing fundamentals. Gifted student learned $X$ by himself, but the material depended on the knowledge of $Y$. Yet, the student lacked proper understanding of $Y$ and so build some bad intuitions, which had to be unlearned before progressing further.
  • Closed-mindness (compared to other gifted students). This one is hard to describe, some signs are:
    • the student thinks there's always a book that explains the area in question;
    • the student specializes in solving toy problems and (difficult) homework, but is unable to handle even simple open or research problems;
    • the student is unable to form his own theories or own interpretations.

There also others, like burnout, but the causes will be more complicated, let's not diverge too much. Even these three trouble me very deeply (especially the last) and it's like some math-contest rat race rather than curious study of the beautiful area that mathematics is.

To me, rediscovering the wheel (if not too often, human lifespan is still limited) is a necessary part of study, so that when the time comes, the kids will be able to do their own original discoveries. Getting all the best concepts laid out in a book, digested, updated and improved over the years doesn't seem like the best idea. Still, perhaps this view is invalid and unfounded?

Question:

Is there any data or research on the effects of access to advanced material early in kid's education?

Given that gifted students are rare, right now even some anecdotal evidence would be great.

What are your experiences in this matter?

More concrete examples:

@Joseph Van Name asked for examples, so here there are.

Over-reliance on the already-learned material:

  • In a geometry class there was a guy that tried to solve every single problem using analytical approach: coordinates, trigonometry, algebra, etc. I admit that he was very good at this, but he didn't learned some techniques we practiced using simpler problems and then wasn't able to solve the more difficult ones which wouldn't yield to his standard method.
  • During programming class there was a group who liked suffix trees too much and some students didn't learned some simpler techniques (here I mostly mean variations on the Knuth-Morris-Pratt algorithm) which were necessary later.
  • There was another guy who would try to solve all the contest inequalities using Muirhead's inequality. He had a huge problem each time the inequality in question was not similar to any mean (e.g. all the numbers begin equal not causing equality).

Missing fundamentals:

  • A person who had learned more advanced probability before university disregarded a few first classes and then assumed that any distribution has a density function. There were later some problems with conditional expected values and their $\sigma$-algebras, but I don't remember it well enough.
  • Many times students who learn asymptotic complexity by themselves don't understand some concepts properly and don't pay enough attention to details. Some common misconceptions are
    • $n \notin O(n^2)$,
    • $n^2 + n \in \omega(n^2)$,
    • $f_i \in O(g)$ implies $\sum_{i=1}^{n} f_i \in O(n\cdot g)$.
  • Frequently students that learned programming without any supervision or guidance from more experienced people have bad style (and programming contests often reinforce that bad style). It is an issue, because it's hard to unlearn and the drawbacks are not severe enough for the student to want to change it by himself. The effect is that the student penalizes himself for a much longer time (one of the reasons why summer internships or involvement in open-source projects is a good idea).
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    $\begingroup$ I too read the question. I'm not against students reading high level books per se. My main concern is the whole child well being including fostering a sound social, moral and behaviour. $\endgroup$ – Karl Mar 30 '15 at 19:22
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    $\begingroup$ dtldarek. Perhaps you should give some examples in the "Over-reliance on the already-learned material" and "Missing fundamentals" questions. $\endgroup$ – Joseph Van Name Mar 31 '15 at 19:38
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    $\begingroup$ @JosephVanName I've added some examples, perhaps you will find them useful. $\endgroup$ – dtldarek Apr 1 '15 at 11:06
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    $\begingroup$ This report on the issue of acceleration vs. enrichment might be of interest: education.lms.ac.uk/wp-content/uploads/2012/02/… My own feeling is that the entire normal curriculum for average students prepares them poorly for higher math, and this effect may be exacerbated with younger students. Really, gifted students should take the time to learn elementary material more in depth than their peers. However, success is often objectively measured in terms such as "I passed such or such higher course", so students may feel some pressure (perhaps... $\endgroup$ – Keith Jun 26 '15 at 20:00
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    $\begingroup$ (cont'd) self-imposed), to move on to higher courses rather than dwelling on more elementary algebra, arithmetic and geometry, even though these subjects can be studied in a way that makes them challenging and interesting. $\endgroup$ – Keith Jun 26 '15 at 20:03
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You asked for anecdotal evidence.

I was a "gifted student". The school told me to teach myself 11th grade math (Trigonometry and Algebra 2) in 9th grade. I never formally learned algebra 1, but I understood it. They gave me a book for 11th grade math and I learned. I don't think it did me any harm - but I do think I understood the concepts and didn't just learn algorithmically. It gave me a lot of self confidence to do this and kept me from being bored in math class. Everyone was delighted when I got a 97 on my 11th grade regents.

In 10th grade we had geometry which I didn't have a strong sense for and so I rejoined my class.

In 11th grade they had me teach myself calculus. When the supervising teacher asked me what I was up to, I said integrals and discovered I was pronouncing it incorrectly. I focused on the AP test and was able to get a 5. In hindsight, I didn't learn the concepts as deeply as I might have from class - perhaps because I was focused on a test?

When I filled out college applications, I wrote about my success with independent study with great pride.

In 12th grade, I was allowed to take college classes and I enjoyed linear algebra and Calculus 3.

Fast forward to my senior year in college and I took a graduate seminar in combinatorics. We had to read and present journal articles. I was delighted to see that my early independent study had trained me to read journal articles. I knew that if I persevered, I would understand them.

If I could do it all again, I would. I believe that independent study was a great opportunity. It taught me to persevere in my learning and gave me confidence. It kept me from being bored. I think it helped that in 10th grade and 12th grade I was back in a classroom that was on my level.

Fast forward to the present I am a gifted math teacher. I have students at the elementary level who like to fast forward using Khan Academy. I don't think it has the same challenge as working through a course, but it keeps them from being bored in class. At times I give these students horizontal enrichment, which is truly what they need.

If you do have such students, I would encourage them, but at the same time I would design intermediate assessments for them that would highlight some of your concerns about how they are using the material. I would also offer material that challenges them without going ahead.

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    $\begingroup$ A gifted math-teacher or a gifted-math teacher? :-) $\endgroup$ – Joseph O'Rourke Jun 26 '15 at 12:24
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    $\begingroup$ @JosephO'Rourke Officially a gifted-math teacher, but I strive to be both! $\endgroup$ – Amy B Jun 26 '15 at 12:48
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    $\begingroup$ Thank you for you answer, I appreciate it very much. As I am not familiar with US math courses, could you describe in some way how the levels of 9th grade and 11th grade compare? There surely is a change in quality between high school and university, but in Europe the difference between grades in high school is only slight. Please don't get me wrong, in no way I would like to diminish your achievement, only that I wouldn't call any of my own high-school books advanced (books for kids indeed are written in a differwnt way). $\endgroup$ – dtldarek Jun 26 '15 at 21:14
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    $\begingroup$ In those days when I was in high school (early 70's), 11th grade included logarithms and trigonometry which weren't touched on at all in 9th grade. I believe the quadratic equations was left for 11th grade as well. Today 9th graders would probably learn some of this material and go on to pre-calculus in 11th grade and the syllabus would be totally different. $\endgroup$ – Amy B Jun 26 '15 at 21:20
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I have a bit of anecdotal evidence.

I was unfortunately not homeschooled, nor did I have a technical childhood; I spent my childhood painting and writing short stories. I was in gifted classes, but I was not seen as a particularly bright student. Due to bullying I looked for alternatives to the local high schools, and ended up applying to university early to get out of town. In 2013, I graduated university at 16. Since then, I've had the full financial support of the Thiel fellowship, which allows for completely self-directed research for those who thrive in such an environment.

I have no formal training in pure math (my bachelors degree is in computational physics and my research was in robotics). I've been teaching myself pure math full time since July 2014. In July, someone asked me if I knew what the Atiyah Singer Index Theorem was. I didn't! I'd never learned topology or anything past numerical analysis.

When I first started, I was quite lost in the literature. I learned from the top down, and the blog posts of John Baez were the only thing on the internet I could understand which discussed the topics I was interested in. I became obsessed, and on the way latched on to trying to understand what elliptic cohomology was -- what concepts led up to it, and what concepts flowed from it -- and how it might lead to a generalization of the Atiyah Singer Index Theorem.

When I first began, I didn't know that I had anyone to talk to. I'd print out a paper by Segal or Jacob Lurie, and underline every concept "x" I didn't understand (which was most of the paper). Then, I'd search "john baez x" and read all posts he had on each concept. I did this for the first 5 months, until I gained enough vocabulary and context to begin additionally reading Lurie's Higher Topos Theory, and Stolz-Teichner's What is an elliptic object?. Then I moved to Berkeley and began sitting in on seminars.

As you might imagine, my knowledge base is woefully incomplete and a bit more "out of order" than usual (learned category theory before set theory, learned what a cyclic group was a few months ago, studied Morava E-theories before basic complex analysis, etc.) This Ravi Vakhil quote resonates with me quite deeply:

Here's a phenomenon I was surprised to find: you'll go to talks, and hear various words, whose definitions you're not so sure about. At some point you'll be able to make a sentence using those words; you won't know what the words mean, but you'll know the sentence is correct. You'll also be able to ask a question using those words. You still won't know what the words mean, but you'll know the question is interesting, and you'll want to know the answer. Then later on, you'll learn what the words mean more precisely, and your sense of how they fit together will make that learning much easier. The reason for this phenomenon is that mathematics is so rich and infinite that it is impossible to learn it systematically, and if you wait to master one topic before moving on to the next, you'll never get anywhere. Instead, you'll have tendrils of knowledge extending far from your comfort zone. Then you can later backfill from these tendrils, and extend your comfort zone; this is much easier to do than learning "forwards". (Caution: this backfilling is necessary. There can be a temptation to learn lots of fancy words and to use them in fancy sentences without being able to say precisely what you mean. You should feel free to do that, but you should always feel a pang of guilt when you do.)

I think this is enough context for me to go over the main issues that I face currently.

The first is that I've raced ahead, and have done so by relying on intuition. It's bittersweet, a bit of a curse, to be a young person who can pick up the structure of complex things quickly by reading. If I am not strong enough, the temptation to keep doing that lures away from making the mistakes and day long fiddling sessions which seem to be useless (i.e. seemingly less productive process of digging deeper and rolling around in the dirt), but are necessary for developing intuition and creative obsession. I have been only recently finding the motivation to go deeper.

I just started writing proofs a month ago when a professor kindly began to send me exercises and then tear my responses apart. My proofs are atrocious and sloppy as I have only recently started writing them.

At the moment, my understanding is not sufficiently primitive and wild for me to feel comfortable and confident in expressing my own ideas clearly. I'm starting to take seriously the disease that commonly affects people of my type:

They race ahead, and don’t learn things precisely, and worse they get into the habit of learning things imprecisely, and even worse never learn what it means to learn something precisely.

The second issue is that I am unfamiliar with the standard undergraduate curriculum (vocabulary and grabbag of examples). This leads to awkward situations. For example, I was having an exciting conversation with a grad student about elliptic cohomology theories, and asked "Pardon, what is the 'characteristic of a field'?" and the previously engaged and kind person became disoriented and extremely suspicious. Similar things have happened many times, though the severity has lessened now that I have learned not to take it personally and to explain my situation.

The third and main issue is that I resisted learning the history and context of math for so long, because I thought that was the best way to gain an original perspective, the best way to learn the art of blundering about in the darkness.

But that's like a writer not reading the classics to be more original and writing the same trope over and over and over. Rather, as a writer you need to reflect on your experiences, let yourself live as your character, ask how they relate to you, ask how they would act in different situations, how they develop. It's like instead of developing an idea for a novel yourself, telling it to others and scribbling down their responses to piece together a story, which is part of it. But this obviously won't work alone, you have to reflect yourself on how to weave the story together. Even if the character is an alien living in a strange world, perhaps a key development can be taken or inspired from another novel, but you can't expect to just copy and paste a part of the novel. You have to rewrite it,adjust it to fit key elements of your character and plot. Reading just one genre won't do either. Perhaps that key snibbet, that phrase that's just right, will come from a child's comedy show, or from a note bolded in an obscure french book.

I have slowly begun to disband my impatience to really appreciate that mathematics really is a narrative -- most of it found in classical books and concepts. But I have met far too many young students who refuse to read the classics at all for fear of "losing time," preferring the slick modern exposition to the fumbling origins of the subject.

The thing to keep in mind is that all these "basics" were once cutting-edge research. And the people who got to the bottom of these matters were not trivial idiots. They were great masters. The more you look at this material in this way, the more you'll find you can tolerate it in good humor, and without fearing its deadening effects on your strange, wild mind.

I think that's enough rambling about myself, hopefully some of it was helpful.

Edit (August 2018):

Shortly after the above post was written, I was accepted into UChicago for a master's program in Maths (the Bridge Program). I struggled mightily, I didn't know the basics of proof writing, and I took many classes multiple times. I loved my reading courses, and my professors were very hard on me. I could not understand why I found some things so difficult, when they were so easy for my fellow students. This allowed me to look upon many of the techniques thought of as "standard" with awe and appreciation. I tried ignoring my math playtime to focus on classes but that made me crumble inside, finally I let myself play (mostly in secret) to stay alive alongside my classes. I graduated with a Masters after much sweat and blood and moved to Northwestern University, where I am now entering my 2nd year in their PhD program. I love Northwestern, it is a great fit for me, as it has other grad students interested in mathematical playtime.

Now that I have had a few years of formal training, and a systematic working through of some key textbooks on my own, my knowledge base is now mostly indistinguishable from another my academic age who took a more "straightforward" path. I still occasionally find impressive and unusual holes in my knowledge base, but it becomes increasingly rare, and I know to bow my head and fill them in.

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    $\begingroup$ Certainly helpful, I appreciate your interest. You describe three issues, but also that you are able to manage them (which is great). In this light, would you change anything in your past if you could (e.g. give your younger self an advice)? Or perhaps this is the best way, because all the deficiencies can be handled when they arise? $\endgroup$ – dtldarek Jul 3 '15 at 20:27
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    $\begingroup$ I'd urge my younger self to read Lakatos Proofs and Refutations, Klein Elementary Mathematics from an Advanced Perspective, and to do many many exercises (rather than just read and pick up keywords). $\endgroup$ – Catherine Ray Jul 3 '15 at 20:39
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    $\begingroup$ One off-topic question, you have mentioned writing (and I assume reading) short stories. As I am always on hunt for texts that could interest bright young minds, do you have any list of fiction that you liked most? To give a few examples: Ra, The Last Ringbearer, Blindsight, Worm or recent …and I show you how deep the rabbit hole goes. $\endgroup$ – dtldarek Jul 4 '15 at 14:21
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    $\begingroup$ A few that immediately come to mind: Microcosmic God, Night on the Galactic Railroad, The Phantom Tollbooth, Tiny Beautiful Things, and Permutation City. $\endgroup$ – Catherine Ray Jul 6 '15 at 4:46
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    $\begingroup$ Hi Catherine I am inspired by your answer and it describes me when I was younger where I would simply "learn" things without knowing them deeply enough. My interests were in physics so I would learn about stuff in quantum field theory, M theory, after having some bare minimum understanding of the basic calculus. But years after I sort of lost my interest due to family problems (having to work to support my family) and now I am in university and I am playing it very safe...a bit too safe in my opinion which involves thoroughly going through a subject using a standard text before moving on $\endgroup$ – Shamisen Expert Jul 14 '15 at 7:20
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To supplement the other U.S.-centric answer: yes, the U.S. standard curriculum through high school is not now and has not been in recent years comparable to several western European or former-easter-bloc education, for a variety of reasons which are irrelevant to the question. But, either way, there are some implicit hypotheses that are (in my opinion) worth being a little skeptical about. First, I am not confident that "the curriculum" is a complete guide to either the structure or content of mathematics, or of the right order to look at things. E.g., neither "acceleration" nor "breadth/depth/enrichment" (in the usual very flimsy sense!) necessarily recommend themselves for very-capable kids. E.g., textbooks, especially contemporary ones in the U.S., of the sort likely to be approved by school boards, are very narrow, very doctrinaire, very authoritarian, and seldom written by anyone who really knows "what will/may happen later".

Even some of the most-popular undergrad texts (and their adherents) seem to accomplish more in the direction of "rigor" and almost peevish attention to detail than giving any sort of breathtaking vistas. Apparently one is not allowed to see those vistas until one has paid some dues?

In particular, returning somewhat to the question, the dangers are not only, or not so much, the possibility of immature response to the material, but that the material itself (whether accelerated-into or widened/deepened in the stereotypical ways) is misleading in several ways.

(Another complication in the U.S. is that the system does not require, and could not currently require, high school math teachers to be acquainted with any substantive mathematics... again, political issues, but nevermind. Nevertheless, this situation makes them almost uniformly completely unprepared to give competent advice even to even-modestly-precocious high school kids.)

One of the most poignant issues is that (certainly in the U.S.) capable kids interested in math get a very warped picture about what it is, or, rather, might be, or, rather, would eventually become, if they had advice about where to look. In particular, mathematics need be neither a "school subject" (with all the authority-tainted issues) nor a "contest subject" (with all the entailed baggage). In particular, if one steps outside those "controlled" situations, one is allowed to read about solution of equations in radical whether or not one understands well enough to "do homework" or "pass exams". The pop-culture mythologies that reinforce "rules" about legal reading of mathematics may or may not be good advice for doing well in classes, in the short term, but ... especially for kids who're not just looking at mathematics because they are told to... this inculcates a very suboptimal attitude and approach.

As a corollary, especially an interested person can often get much insight about the sense and intent of many details by seeing what happens later, how they're used, how they interact with other ideas. Most standard textbooks are really bad at telling any of that, and, in fact, often must particularly avoid any whiff of connection to other things, because then it becomes more awkward to "control" the curriculum, etc. Things are not so interchangeable, ... don't have clear prerequisites, heaven forbid!

(These remarks in the context of having worked at pretty decent universities for 40+ years, having had the privilege of having some very good students, both undergrad and grad, along the way, and that sort of thing... but also having been an accidentally slightly precocious kid back in pre-internet days when the public library was about as good as it got, simultaneous with U.S. high schools being much more authoritarian and rule-ish. So, for example, a person might accidentally hear about "calculus", and see amazing things that it can do. Or complex numbers, DeMoivre's identities, complex analysis! ... amazing stuff. Galois theory... solving cubics in radicals... algebraic number theory... Godel's theorems... set theory... all this stuff well-known to professional mathematicians, but pushed off into the hazy distance in the standard curriculum, and too often completely unknown to U.S. high school math teachers. Meanwhile, in those days, high school math was memorization of various goofy formulas for things, derived by as-elementary means as possible, even in scenarios where calculus would trivialize things. Elaborate trig identities... For a long time, I was unaware that there could be any sort of profession of "mathematician", since it appeared that as a school subject it was just awful, although engineers and actuaries "did some math". I could not get any straight answers about "where" the kind of mathematics in those books in the public library might be manifest. It was only by accident some years later that it became painfully clear that the world of even "enhanced" high school math was/is basically unrelated to anything serious... which I think is the relevant question for truly talented kids.)

Conclusion: for really able kids, get outside the curriculum entirely. Don't look at the curriculum for guidance. In the U.S., this would, in my estimation, refer to high school and standard undergraduate programs, which are both highly authoritarian and very limited in what is expressed.

(The fact that "getting outside the curriculum" may short-term appear disadvantageous is counter-balanced by the anecdotally-documentable fact that less authority-driven reflection on important mathematical ideas pays off both long-term and even middle-term.)

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    $\begingroup$ Thank you for your insights, I find them higly relevant. For the record, I've learned the applied side of integrals and differential equations while still in primary school (5-6th grade?): I wanted to code a game similar to lunar lander and my father said we will do things right, which involved writing a bunch of physics equations and solving them numerically. It took many days, but it was great and gave me valuable intuitions I have used innumerable many times. Cont. $\endgroup$ – dtldarek Jun 29 '15 at 21:17
  • $\begingroup$ Cont. On the other hand, I haven't been introduced to ODEs rigorously until the university. Although I think I could grasp that somewhere around high school, I doubt it would be beneficial, if not even harmful. And, certainly, at that time, at university-level of detail, I would find these concepts utterly boring. Fin $\endgroup$ – dtldarek Jun 29 '15 at 21:18
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    $\begingroup$ I think being bored as a reaction is completely legitimate. Either one decides reasonably-enough on one's own that something is important, or is persuaded (not bullied, not threatened) to believe it. I was very glad that I was allowed to skip over ("test out of") the grimmest parts of the standard undergrad math curriculum... $\endgroup$ – paul garrett Jun 29 '15 at 22:32
  • $\begingroup$ I am not qualified to discuss university math, I myself changed my major after realizing that analysis was too complex for me to understand. But I do have an opinion regarding American middle and high school math. As mentioned in the answer, it is quite bad, and getting worse. The progression of Algebra 1 -> Geometry -> Algebra 2 has been replaced in many disctricts with intergrated approach, and the only textbook that teach in integrated fashion are the NCTM creations from the 1990s, with a thin coat of Common Core paint, like Core Plus course. This is a rather horrible course... (cont) $\endgroup$ – Rusty Core Sep 6 '18 at 23:14
  • $\begingroup$ ...disorganized, with little theory scattered around and with push onto usage of graphing calculator. Even if one were able to successfully study this course, one would be able to get pre-calc in grade 12, don't even think of calc of whatever else. Therefore, accelerating math in HS makes sense for several reasons: getting through this swamp quicker; getting to juicier stuff like AP Calc faster; potentially using more rigourous and sane textbooks. AP Calc or AP Stats are valuable both as extra points for college applications, as well as a college credit. $\endgroup$ – Rusty Core Sep 6 '18 at 23:14
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Sorry if this sounds simple, but I think the biggest consideration is efficiency (maybe a bit related to the missing Y issue). If you study books that are too hard, it may take you longer than a more progressive approach would get you (even including eventually using the tougher text).

In addition, the danger of giving up is not to be trivialized. I think there is a balance between the joy of working through difficulty and the value of a kinder, progressive training. In addition, even if the book is "easy" one can still get joy from it by covering it faster.

Also, I find way more value in topics I master than those I don't. Courses I got A+ in stick with me for decades. B or C classes (I went to a school that still gives those) never "sunk in". So if you master an easy text, you may have more benefit than if you just have partial success in a tougher one.

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I was introduced to the concept of infinity early so I formed the intuition of the hyperreal number system. I think I also remember being taught in School that 0.999... = 1 and not understanding why that was true. I think that might have been after the time I was introduced to the concept of infinity and that might have been the reason. I was also told that $1 \div \infty = 0$ outside of school but my intuition told be that $1 \div \infty \neq 0$ which is how the hyperreal system really works. I think I may have also thought that only the notations that eventually terminate whether at finite or infinite position represent a number similar to the method of thinking shown at https://www.youtube.com/watch?v=wsOXvQn3JuE as a result of being introduced to infinity early and from that assumption deduced that $\frac{1}{6} = \frac{1}{3} \div 2 = 0.333...333 \div 2 = 0.1666...6665$ because I had already figured out on my own how to divide by 2 in decimal. Without realizing it, my assumptions as a whole were contradictory so it can in fact be shown that $\frac{1}{6} = 0.1666...6665 \neq \frac{1}{6}$. It may have come from the Burali-Forti paradox of Naive set theory.

Much later, I thought I finally figured out a way to construct the real numbers properly from the dyadic rationals and understood that the real number system is a different system where there are no infinite or infinitesimal numbers. I later discoved Cantor's paradox by myself. Even later, I discovered by myself that I cannot prove the axiom of choice. It turns out that without realizing it, I didn't really figure out how to construct the real numbers because I had also earlier thought of my own definition of a natural number independently which turned out to be the same as the definition on page 25 of the book A first course in real analysis which said an inductive set is a subset of $\mathbb{R}$ such that 0 is in the subset and for every real number in the subset, that number plus 1 is also in the subset and a natural number is a real number that every inductive set contains. Now I realize that you can construct the natural numbers as finite ordinal numbers then construct the integers then construct the dyadic rationals then construct the real numbers.

I was 21 at the time I discovered on my own that I can't figure out how to prove the axiom of choice and I feel like after the time I discovered that, I got helped even more by discovering it on my own at that age than I would have by being told at a young age why it wasn't provable. I think by brain was muture enough that it could handle the change and I acutally liked breaking by old habits and learning how to adapt to think in a totally different way, unlike the Pythagoreans who hated the discovery that irrational numbers exist. I'm glad I didn't get told that at a younger age because then otherwise, my brain might have missed out on breaking its old habits at an older age and becoming even better as a result. https://www.inc.com/bill-murphy-jr/science-says-were-sending-our-kids-to-school-much-too-early-and-that-can-hurt-th.html seems to be a sign that my brain may have actually benefitted from discovering the unprovability of the axiom of choice at the age of 21.

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