# Thread: Mathematical No-Man's Land b/t Intuition & Comprehension???

1. ## Mathematical No-Man's Land b/t Intuition & Comprehension???

Hi. I've had a question about Mathematical Aptitude for some time but, until now, had been hesitant to try and express it. Here it goes...

For some time now, I've been wanting to engage in an in-depth study of Mathematics. In fact, I've begun such an endeavor on several occasions– only the pursuit never lasted for very long, maybe a week or two, before I get distracted by other things. Still, every time I do pick up one of those Math books– something happens that I always find strange: While I find myself rarely fully grasping the more advanced Math formulas and equations, in some strange way I feel like I can intuit its meaning. That is to say, while legitimate comprehension of the Mathematics lies outside my grasp– outside of my ability to actually DO the Math or EXPLAIN it– somehow, despite this reality, I still seem to GET the math. It makes SENSE. It's just... fuzzy? This experience is all the more heightened on those occasions where I'm actually actively trying to do the Math. I'll be writing the problem out, and struggling with it but, at the same exact time, I'll be looking at it and be knowing that I do get it, I just can't DO it. Sometimes I'll even test myself by predicting the next step in the solving of a problem and I'll be correct– but I can't explain how I just know that's the next step. Again: I INTUIT it– but I can't DO it. The best analogy I could probably offer is the experience of a word being on the tip of the tongue: You KNOW you KNOW the word– you just can't SAY it. As if there's some block between your brain and your tongue. This sort of cognitive phenomena happens to me nearly every time I crack open a Math book and flip to sections that are clearly out of my depth.

What made this experience even stranger for me was an experience I just recently had with a Math text. In endeavoring to explain the rudiments of Mathematical language, the author of the text offers the following phrase: 'For every thing, if that thing is a cloud then there is a thing such that that thing is a silver lining and the first thing has the second thing.' He then goes on to convert that phrase into Mathematical language. But the illuminating part for me was the author pointing out that phrases like 'for every thing' and 'there exists a thing' are called quantifier phrases. This stood out to me because, whilst I am definitely no Mathematician, I excel in the department of Natural Language (I'm a writer) and, as it turns out, I frequently employ quantifier phrases in my writing. In that same vein, the fundamentals of syntax [which are far more strict in the language of Mathematics than English] also come quite naturally to me when I write. In essence, these things only further this sense I have that, buried somewhere inside the haze of my decidedly qualitative mind, lies a sufficiently quantitative one.

In any case, I say all this to pose a few questions: 1) Has anyone else experienced this sort of cognitive phenomenon? 2) For any one who has had this experience or is in any way familiar with it– do you know of any way to perhaps cut through the fuzziness and sharpen the crude Mathematical aptitude buried within? I ask this second question because I often have this feeling that I may just not be approaching the Math the way someone with my particular mind is best designed to approach it. That if I just attacked Math from a slightly different angle, I might start to see progress at a considerably quicker rate.

Sorry for the long post. I just wanted to communicate the experience as clearly as I could. Thank you for reading.

P.S. - I should mention that I've also dabbled in C++ [which, while technically designated a formal language, tends to fall in between the strict formality of Pure Mathematics and the high malleability of Natural Language] and have had the same experience, albeit in a less severe way.

2. Disclaimer: I've been enjoying my wine, so keep that in mind below.

Originally Posted by workinprogress
… 1) Has anyone else experienced this sort of cognitive phenomenon?
Regularly!

And, the deeper I think about the phenomenon itself, the sooner I end up in philosophy and/or metaphysics.

Albert Einstein has described some of his breakthroughs as seemingly having come from an outside source, while in a trance-like state of thought. Certain aspects of quantum theory (in one interpretation) suggest that true information as data exist on the surface of the universe, while everything we see or think we know is an inward-projected copy of that "truth". Far-out stuff. We don't really know where the source of thought lies, but maybe what you described using the on-the-tip-of-the-tongue analogy is just a normal attribute of how ideas "get into" our mind. That is, maybe there's no way around it. Be thankful! Some people have strong intuition; some others do not have much. What exactly is intuition? That's not an easy question to answer. Gosh, I could branch off in many directions here, so I'd better stop.

2) For any one who has had this experience or is in any way familiar with it– do you know of any way to perhaps cut through the fuzziness and sharpen the crude Mathematical aptitude buried within?
Time and effort.

I ask this second question because I often have this feeling that I may just not be approaching the Math the way someone with my particular mind is best designed to approach it. That if I just attacked Math from a slightly different angle, I might start to see progress at a considerably quicker rate …
First, I strongly believe that anybody can achieve success in mathematical endeavors; science has disproven the myth that some people have "math brains" and others don't. The brain has amazing plasticity; what it needs is experiences, in order to "recognize" and encode patterns.

Second, math is a subject of many accumulated, interconnected ideas. With more and more effort comes more and more experience, and the easier the whole endeavor becomes (eg: one thing leads to another because they have the same pattern, one thing can be accomplished in the same way as a different thing because they're connected, new stuff can be interpreted in terms of previous stuff). From my point of view, continued exposure to many mathematical concepts and patterns (some old, some new) is what cuts through the fuzziness -- that is, doing math. You'll never reach a point where the fuzziness is gone; it just shifts to newer ideas for which you don't yet have sufficient experience to see "clearly".

3. Originally Posted by workinprogress
Hi. I've had a question about Mathematical Aptitude for some time but, until now, had been hesitant to try and express it. Here it goes...

For some time now, I've been wanting to engage in an in-depth study of Mathematics. In fact, I've begun such an endeavor on several occasions– only the pursuit never lasted for very long, maybe a week or two, before I get distracted by other things. Still, every time I do pick up one of those Math books– something happens that I always find strange: While I find myself rarely fully grasping the more advanced Math formulas and equations, in some strange way I feel like I can intuit its meaning. That is to say, while legitimate comprehension of the Mathematics lies outside my grasp– outside of my ability to actually DO the Math or EXPLAIN it– somehow, despite this reality, I still seem to GET the math. It makes SENSE. It's just... fuzzy? This experience is all the more heightened on those occasions where I'm actually actively trying to do the Math. I'll be writing the problem out, and struggling with it but, at the same exact time, I'll be looking at it and be knowing that I do get it, I just can't DO it. Sometimes I'll even test myself by predicting the next step in the solving of a problem and I'll be correct– but I can't explain how I just know that's the next step. Again: I INTUIT it– but I can't DO it. The best analogy I could probably offer is the experience of a word being on the tip of the tongue: You KNOW you KNOW the word– you just can't SAY it. As if there's some block between your brain and your tongue. This sort of cognitive phenomena happens to me nearly every time I crack open a Math book and flip to sections that are clearly out of my depth.
I am going to start with a bit of biography: I am not a mathematician despite tutoring the subject here and face to face. My academic training was in European history and languages; my professional career for decades has been in finance though I started in the early 60's as a FORTRAN programmer. Fortunately, I went to a great prep school that essentially taught only languages, math, and science. It was sort of a post-doc in the three R's. So I know nothing about the current frontiers of mathematics, but I know a few basic topics in mathematics well.

It is true that mathematical NOTATION is a language, one that has been carefully tooled to be concise and unambiguous. $\exists\ x$ can be translated into English as "there exists a thing temporarily named x." So one reason everything seems to make a "fuzzy" kind of sense is that it is not hard to grasp the "grammar" of the language.

What is difficult is that the concepts are very abstract. Arithmetic is about numbers. But try to define exactly what a number is. What is the distinction behind a number and a numeral? What is the distinction behind an ordinal and a cardinal number?

Furthermore, the principles you are to learn are general principles. General principles about abstract concepts only loosely understood cannot be applied. You get the demonstration, but cannot replicate it. This is why learning mathematics requires incredible time in drills. You must encounter enough examples to perceive what is being generalized. It is also why teaching mathematics is hard. Once something has become clear, you can remember that something was once obscure without being able to remember exactly why it was ever obscure. "Si j'avais su ce que je sais" is the refrain of a poem about regret for lost innocence, but it works for lost ignorance as well. I confront this problem every time I try to get some bewildered kid to understand what a function is.

EDIT: Also mathematics is both logically and historically progressive. (Morris Klein strongly argued that it was psychologically better to follow the historical path in teaching rather than the logical path.) in any case, you cannot "flip" around. Newton, an admittedly pathological case, claimed to have started rereading Descartes' Geometry from the beginning each time he got stuck because that presumably meant that he had not fully understood something at an unknown point in the previous text. (The 17th century had a rather exaggerated opinion of Descartes.) It is the progressive nature of mathematics that explains why you cannot "flip" around and expect more than an understanding too vague to be applied.

4. Originally Posted by mmm4444bot
Disclaimer: I've been enjoying my wine, so keep that in mind below.

Regularly!

And, the deeper I think about the phenomenon itself, the sooner I end up in philosophy and/or metaphysics.

Albert Einstein has described some of his breakthroughs in thought as having come from an outside source, while in a trance-like state of thought. Certain aspects of quantum theory describe true information as data existing on the surface of the universe, while everything we think we know is an inward-projected copy of that "truth". Far-out stuff. We don't really know where the source of thought lies, but maybe the word-on-the-tip-of-the-tongue effect is just an attribute of an unknown process involving delays from source to destination. That is, maybe there's no way around it.

Gosh, I could branch off in many directions here, so I'd better stop.

Time and effort.

First, I strongly believe that anybody can achieve success in mathematical endeavors; science has disproven the myth that some people have "math brains" and others don't. The brain has amazing plasticity; what it needs is experiences, to recognize and encode patterns.

Second, math is a subject of many interconnected ideas. With more and more effort comes more and more experience, and the easier the whole endeavor becomes (eg: one thing leads to another, one thing can be accomplished in the same way as a different thing because they are connected). From my point of view, continued exposure to many mathematical concepts and patterns (some old, some new) is what cuts through the fuzziness -- that is, doing math. You'll never reach a point where the fuzziness is gone; it just shifts to newer ideas for which you don't yet have sufficient experience to see "clearly".

5. Originally Posted by JeffM
I am going to start with a bit of biography: I am not a mathematician despite tutoring the subject here and face to face. My academic training was in European history and languages; my professional career for decades has been in finance though I started in the early 60's as a FORTRAN programmer. Fortunately, I went to a great prep school that essentially taught only languages, math, and science. It was sort of a post-doc in the three R's. So I know nothing about the current frontiers of mathematics, but I know a few basic topics in mathematics well.

It is true that mathematical NOTATION is a language, one that has been carefully tooled to be concise and unambiguous. $\exists\ x$ can be translated into English as "there exists a thing temporarily named x." So one reason everything seems to make a "fuzzy" kind of sense is that it is not hard to grasp the "grammar" of the language.

What is difficult is that the concepts are very abstract. Arithmetic is about numbers. But try to define exactly what a number is. What is the distinction behind a number and a numeral? What is the distinction behind an ordinal and a cardinal number?

Furthermore, the principles you are to learn are general principles. General principles about abstract concepts only loosely understood cannot be applied. You get the demonstration, but cannot replicate it. This is why learning mathematics requires incredible time in drills. You must encounter enough examples to perceive what is being generalized. It is also why teaching mathematics is hard. Once something has become clear, you can remember that something was once obscure without being able to remember exactly why it was ever obscure. "Si j'avais su ce que je sais" is the refrain of a poem about regret for lost innocence, but it works for lost ignorance as well. I confront this problem every time I try to get some bewildered kid to understand what a function is.

EDIT: Also mathematics is both logically and historically progressive. (Morris Klein strongly argued that it was psychologically better to follow the historical path in teaching rather than the logical path.) in any case, you cannot "flip" around. Newton, an admittedly pathological case, claimed to have started rereading Descartes' Geometry from the beginning each time he got stuck because that presumably meant that he had not fully understood something at an unknown point in the previous text. (The 17th century had a rather exaggerated opinion of Descartes.) It is the progressive nature of mathematics that explains why you cannot "flip" around and expect more than an understanding too vague to be applied.
Another great reply. And one I also fully agree with. Thank you. The abstraction of Math does seem to be what throws many people off. Although it can be taxing, it's exactly why the pursuit of Mathematics is valuable to me. Abstraction is such a powerful tool (both in science and the arts) to have in the arsenal. As I stated in my previous post, I'm a longtime student of metaphysics, which could be defined as first-order philosophical abstraction, and it really does enable you to make sense of seemingly disparate worldviews and reconcile them such that you come to a much deeper understanding of 'all' things (by way of their relationships to one another). I actually think that previous sentence has something to do with my intuition when it comes to Mathematics. I look at all that foreign notation and, yet, somehow, my brain is associating it with things I am familiar with (analogy). An experience not unlike seeing someone on the street you swear you've met before a long time ago but you just can't place where or when. Also: Morris Kline. He is one of the tops for me. I actually consider him to be the best teachers of Mathematics I've ever come across. Anytime someone expresses discomfort or fear of math I recommend his book 'Mathematics For The Non-Mathematician'. Such a strong and clear treatise on Math History, Logic (Deductive Reasoning) and axioms– which he integrates seamlessly throughout the work, building up from rudimentary concepts in ways that make you say, 'Oh, THAT'S why THAT happens' again and again'. Lastly, great point about 'flipping around' (something I admittedly have done often). Math, for most, is really something that should be studied from the basics– at least early on. Ties right into the sound advice that it really just comes down to doing the drills consistently. Thanks a million.

6. Seeing as how you two are clearly knowledgeable, and I have your attention for the moment– if it's not too much trouble I have one more question for the both of you: Can you each recommend your top choices for books with which to begin an earnest study of Math? Two of the choices I've seen recommended quite often are 'Principles Of Mathematics' by Allendoerfer/Oakley and 'Basic Math' by Lang. How do you two feel about those? And are there any others you'd recommend that fall into that 'for beginners but pure' category?

7. Originally Posted by Denis
I now know everything I need to know. Not just about Math. About Life.

8. As I said, I am not a mathematician. I merely tutor in the basics that I learned decades ago and used often enough since then to remember well. So I cannot advise on books. I suspect that the translations of Bourbaki represent one of the purest and most rigorous developments of mathematics from the ground up, but I do not know how accessible they are to those who are not trained mathematicians. In fact, I do not know whether they have been translated into English. And they may have been superseded.

And I agree with you about Morris Kline. The third volume of Kasner and Newman has some fabulous essays on mathematics, particularly the essay by Gasking and the two essays by Norman Campbell.

9. Originally Posted by JeffM
As I said, I am not a mathematician. I merely tutor in the basics that I learned decades ago and used often enough since then to remember well. So I cannot advise on books. I suspect that the translations of Bourbaki represent one of the purest and most rigorous developments of mathematics from the ground up, but I do not know how accessible they are to those who are not trained mathematicians. In fact, I do not know whether they have been translated into English. And they may have been superseded.

And I agree with you about Morris Kline. The third volume of Kasner and Newman has some fabulous essays on mathematics, particularly the essay by Gasking and the two essays by Norman Campbell.
Word. All good. Thanks a mil.

10. Originally Posted by workinprogress
… top choices for books with which to begin an earnest study of Math? … [which] fall into that 'for beginners but pure' category?
I've never done an earnest study of math, other than in school. Are you asking about undergraduate texts? (I'm not sure what you mean by 'pure').

As the Lang book comes highly-recommended, I'd just start there. It usually doesn't take long, to discover whether you can follow an author or not. You could download some free introductory texts at middle/high school level (i.e., through intermediate algebra) and perhaps a trigonometry text. That way, if you struggle with one of Lang's presentations, you can concurrently look up the same topic using the index in other texts.

I have this one, on my desktop, for use as a quick reference (it begins with intermediate algebra, with some beginning algebra review, and covers some precalculus topics, as well as trigonometry).

I also use this site, when looking for tutoring material (it begins at the grade school level).

Also, I'd like to note that I edited my first post; I had incorrectly paraphrased Einstein, and I modified the rest of that paragraph, as well. Cheers

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