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I have recently started to develop my mathematical intuition. In the past I saw math as a mere game of symbol manipulation, whosoever was able to see patterns and cram formulas and apply them upon those symbols won. However,I now feel that the truth is contrary to the aforementioned,this is because I see in each new chapter that I study in calculus or linear algebra, there is an inherent geometrical intuition that can be easily visualized. At other instances there may not be a geometric interpretation, however there are ways of understanding why the proved results are true. But there is still a void of intuition in the most basic algebra that I am encountering. The following link should demonstrate what algebra I am talking about: http://link.springer.com/chapter/10.1007/978-0-8176-4549-6_1#page-2

Another example further in this text: Finding the roots of the equation: (xy − 7)^2 = x^2 + y^2

Although, the aforementioned can be easily solved after looking at solved examples, however, for each step my motivation is to basically juggle the symbols until I find a pattern I am looking for. What that pattern represents and the symbol juggling itself seems like an opaque layer to me? Please, can anybody give me suggestion to how I can intuitively plan out my steps doing basic algebra. Is there a way of doing math on an intuitive basis from the bottom up? Preferably, in a visual way?

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    $\begingroup$ The link goes to a page where one can buy access to the text. It would be better if you gave the example directly, instead of this link. $\endgroup$
    – Per Erik Manne
    Commented Nov 15, 2015 at 21:45
  • $\begingroup$ @PerManne Let p and q be two primes. Solve in positive integers the equation: 1/x + 1/y = 1/pq $\endgroup$
    – fahad aijaz
    Commented Nov 15, 2015 at 22:23
  • $\begingroup$ This diophantine equation seems like something much more advanced and more difficult than the other example you gave (involving a quadratic equation). $\endgroup$
    – littleO
    Commented Nov 15, 2015 at 23:18

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This is not really an answer, but it was getting too long to be a comment.

Mathematics draws much of its power from deep, sometimes mysterious dualities between geometry and algebra, so I do not think there is any way to understand the relationship between geometric intuition and symbol manipulation in general.

Diophantine equations are one of the least geometrically intuitive areas of mathematics, though it is certainly worth mentioning that geometric insights (albeit very difficult ones) are behind some of the deepest results in the field, like Fermat's Last Theorem.

One can sometimes make arguments from intuitive principles that give partial answers. To take your example of $(xy-7)^2 = x^2 + y^2$, one can observe immediately that this is the intersection in the plane of two curves, one of degree $2$ and one of degree $4$, that have no components in common. By Bézout's theorem, there are at most $8$ intersection points (in fact, we can conclude that there are exactly $8$ solutions, if we know how to count them).

This tells us that the integer solutions, in particular, are a finite, possibly empty set. But it doesn't really give a good a priori method for finding them. That requires some symbol pushing.

To some extent, we can gain a lot of intuition for pushing symbols around. But this can be a lifelong process, and I think any mathematician will tell you that there are advantages to being able to manipulate equations without completely understanding what you are doing. There is some truth to the famous quotation that mathematics is about getting used to things, and for myself I can say that diligent practice has been far more valuable than theory when it comes to developing comfort with difficult problems.

The theory and intuition comes eventually, and with specific examples you may get little boosts to your insight here and there from helpful mentors, but there is no singular approach that will allow you to "plan".

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    $\begingroup$ A great insightful answer! Although I only have high school level knowledge of mathematics, I am of the opinion, "If You Can Specify It, You Can Analyze It" (Flajolet), once analyzed, hacking the problem and after getting a feel for it, the problem becomes significantly easier. For e.g, I believe your insight regarding the 8 intersection points definitely makes you more capable than me solving such a Diophantine equation, as you will be able to grasp a more detailed curve as opposed to my visualization of the same curve. This certainly leads to better chances in solving the problem at hand. $\endgroup$
    – fahad aijaz
    Commented Nov 15, 2015 at 22:45
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    $\begingroup$ @fahadaijaz The problem is that these insights often come long, long after a person has learned how to solve such problems the non-insightful way. Diophantine equations in particular are so hard precisely because they frequently have infinitely many solutions over $\mathbb{R}$ or $\mathbb{C}$, but only finitely many over $\mathbb{Z}$ or $\mathbb{Q}$. $\endgroup$
    – Andrew Dudzik
    Commented Nov 15, 2015 at 22:53
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    $\begingroup$ @fahadaijaz For example, the geometry required to understand, in principle, why $x^2 + y^2 = z^2$ should have infinitely many integer solutions, while $x^3 + y^3 = z^3$ should have only finitely many, is research-level mathematics, while undergraduates can understand the symbolic manipulation. There are many geometric insights along the way, but it is often not possible to start with them. $\endgroup$
    – Andrew Dudzik
    Commented Nov 15, 2015 at 22:53
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    $\begingroup$ @fahadaijaz I don't think that there is a single answer to this question. In general, I don't think that geometrical intuition can "replace" algebra, I think that they are two very different sides of a coin, and much of mathematics arises in the interaction between the two. The closest I know to machinery that can somewhat automate these dualities is topos theory, e.g. Toposes as bridges. $\endgroup$
    – Andrew Dudzik
    Commented Nov 15, 2015 at 22:57
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    $\begingroup$ @fahadaijaz Note that algebraic geometry in general (I don't know much about Clifford algebras) is a very rich area for learning about connections between algebra and geometry. For example, it is because of algebraic geometry that we can understand certain kinds of number theory geometrically, i.e. we can think of the integers as the set of functions on a curve whose points are the primes, and this turns out to be useful for many things. This is all extremely advanced stuff at a high school level, but could be motivation for learning some abstract algebra in the short term. $\endgroup$
    – Andrew Dudzik
    Commented Nov 15, 2015 at 23:03
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Practice.

I might say that the general problem-solving sequence is (per Polya): (1) Read a natural-language problem carefully, (2) Translate to math equation(s); (3) Solve the equation(s); (4) Translate back to natural language and check for reasonability.

Now the truth is that in step #3 you usually do want to be manipulating symbols directly. There is usually some step-by-step algorithm or strategy for a given field of problems; and the whole point of the symbolic writing is that this becomes the most concise and fastest way of solving such a problem. Alfred North Whitehead wrote:

"By the aid of symbolism, we can make transitions in reasoning almost mechanically, by the eye, which otherwise would call into play the higher faculties of the brain. [...] It is a profoundly erroneous truism, repeated by all copy-books and by eminent people when they are making speeches, that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilisation advances by extending the number of important operations which we can perform without thinking about them. Operations of thought are like cavalry charges in a battle — they are strictly limited in number, they require fresh horses, and must only be made at decisive moments."

But! Don't forget to also practice at the end of the symbolic process translating back to natural language and double-checking if the meaning is reasonable (which can be by substitution, estimation, and/or quick graphing). Failing to do that last part does leave a person with a rather empty and brittle set of symbolic-pushing skills.

As usual, the answer is: "You need both".

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“This is not really an answer, but it was getting too long to be a comment.”

I see in each new chapter that I study in calculus or linear algebra, there is an inherent geometrical intuition that can be easily visualized

…And you expect it to generalize to the whole mathematics. Unfortunately, it does not. For example, what you saw in linear algebra does not generalize to algebra. Everything can be proved by symbolic manipulation, but different visual tools are needed for different topics in mathematics, and sometimes there are no visual tool at all, and we have no other choice than to do without it. Hence visual tools are supplements, not a foundation. Finding them is hit-or-miss, I would say it is like collector’s job. ☺

As this answer points out, we can gain intuition about manipulating symbols by practice, I mean, what rule will likely lead to a useful result. Then we can develop visual tools for manipulating symbols. It is the approach which I use. Diagrams in category theory are a famous example, and value-relation diagrams are similar and suitable for more elementary thinking about points, sets, and relations.

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