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Abstract Math Explained: How to Use Abstract Mathematics

 You don't want a problem that is too easy, that is too routine. You always want problems that are just barely out of reach of the known technologies. One analogy that many mathematicians use is to liken mathematical problem solving to climbing. In fact, many mathematicians are rock climbers. I myself am not, but many of my friends are. And actually, rock climbers, they use some of the same language that mathematicians use. A difficult cliff is called a problem. And when you climb the cliff, you solve the problem. If you want to get from the bottom of a cliff to the top of a cliff and you know it's 30 feet high, you can't just go directly up there. That's-- that's not feasible. But you want to find some handhold or some ledge which is more feasible, just almost within reach. You just need to stretch a little bit. And then once you have that point, you identify the next step, which is-- gets you closer to your goal, but one which is just again, just barely within reach. You know, even the best rock climber, there will be certain cliff faces that are just not climbable, even with the best tools and the best physical training. And that's okay. In mathematics, it's okay if you can just get some new progress that is just a little bit above what previous climbers were able to-- to reach. And then the next person-- maybe someone younger and more energetic, or just someone with a different mindset than you-- can go and take the next step. It doesn't matter if you fail nine times. If you make partial success one time and you can explain that success to someone else, you can take it to the next level. That's-- that's-- that's how mathematics proceeds.

Abstract math, a subgroup of mathematics, encompasses the constructions of math problems, formats, and equations to further deepen math concepts for practical use in real-world scenarios. 

Abstract math is a higher-level field of study that can include abstract algebra, geometry, calculus, and other [abstract concepts](https://www.masterclass.com/articles/abstract-thinking). Abstract math is useful because it connects different parts of math. For example, results in a given area can lead to conjectures in another, and mathematicians might generalize and apply patterns from one mathematical object to other fields.
Abstract math is an ever-evolving field of study in which mathematicians refine and create new concepts within math. Consider the development of these different areas of abstract math:

- __Euclidean math__: Attributed to the ancient Greek mathematician Euclid, Euclidean math is perhaps the earliest documentation of abstract math, which noted during ancient times the axioms of plane geometry. 
- __Cartesian coordinates__: In the seventeenth century, [René Descartes](https://www.masterclass.com/articles/rene-descartes-life-and-philosophy) introduced Cartesian coordinates, progressing the use of complex numbers and analytic geometry.
- __Geometry__: European mathematicians such as Carl Friedrich Gauss, János Bolyai, and Bernhard Riemann advanced the field in the 1700s and 1800s. In the nineteenth century, experts generalized geometry even further through the application of n dimensions, projective geometry, and affine geometry.
In mathematical studies and high school math classes, students learn about matrices, linear algebra, topology, polynomials, and other mathematical objects. This helps them understand geometric and algebraic structures to solve theoretical or real-world problems. Abstract math problems might include finding the area of a plot of land during a [land survey](https://www.masterclass.com/articles/land-survey-guide), solving for the mean [price of goods](https://www.masterclass.com/articles/cost-of-goods-sold-explained) to match consumer spending habits, or visualizing symmetric lines for roadways as a part of [urban planning](https://www.masterclass.com/articles/urban-planner-guide).

Australian mathematician [Terence Tao](https://www.masterclass.com/classes/terence-tao-teaches-mathematical-thinking) says, “There is a perception that mathematics is some sort of sorcery.” Abstract math, however, is a highly creative tool that can unlock new modes of thought. “It's a way of thinking,” he says. In collaboration with mathematician Ben Green, Terence used creative thinking to [solve a problem](https://www.masterclass.com/articles/mathematical-problem-solving) related to the Twin Prime Conjecture. The pair proved it’s possible to always find a progression of prime numbers of any length and of equal spacing in the infinity of integers.
Get the [MasterClass Annual Membership](https://www.masterclass.com) for exclusive access to video lessons taught by science luminaries, including Terence Tao, Bill Nye, Neil deGrasse Tyson, Chris Hadfield, Jane Goodall, and more.
Abstract mathematics concerns theorems, algorithms, and other mathematical concepts and broad applications for practical use in the real world. Learn about the history of abstract mathematics.

Abstract mathematics concerns theorems, algorithms, and other mathematical concepts and broad applications for practical use in the real world. Learn about the history of abstract mathematics.

Abstract math, a subgroup of mathematics, encompasses the constructions of math problems, formats, and equations to further deepen math concepts for practical use in real-world scenarios. Abstract math is a higher-level field of study that can include abstract algebra, geometry, calculus, and other <a href='https://www.masterclass.com/articles/abstract-thinking' target='_self' class='mc-text--link' data-mc-segment-trigger='true' data-mc-segment-event-type='Article Page Click' data-event-obj='{"textTag":"p","href":"https://www.masterclass.com/articles/abstract-thinking","linkCopy":"abstract concepts","category":null}'>abstract concepts</a>. Abstract math is useful because it connects different parts of math. For example, results in a given area can lead to conjectures in another, and mathematicians might generalize and apply patterns from one mathematical object to other fields.

What Is Abstract Math?

Abstract math is an ever-evolving field of study in which mathematicians refine and create new concepts within math. Consider the development of these different areas of abstract math:<ul class="mc-m-4 list-styled"><li class="">Euclidean math: Attributed to the ancient Greek mathematician Euclid, Euclidean math is perhaps the earliest documentation of abstract math, which noted during ancient times the axioms of plane geometry. </li><li class="">Cartesian coordinates: In the seventeenth century, <a href='https://www.masterclass.com/articles/rene-descartes-life-and-philosophy' target='_self' class='mc-text--link' data-mc-segment-trigger='true' data-mc-segment-event-type='Article Page Click' data-event-obj='{"textTag":"p","href":"https://www.masterclass.com/articles/rene-descartes-life-and-philosophy","linkCopy":"René Descartes","category":null}'>René Descartes</a> introduced Cartesian coordinates, progressing the use of complex numbers and analytic geometry. </li><li class="">Geometry: European mathematicians such as Carl Friedrich Gauss, János Bolyai, and Bernhard Riemann advanced the field in the 1700s and 1800s. In the nineteenth century, experts generalized geometry even further through the application of n dimensions, projective geometry, and affine geometry. </li></ul>

A Brief History of Abstract Mathematics

In mathematical studies and high school math classes, students learn about matrices, linear algebra, topology, polynomials, and other mathematical objects. This helps them understand geometric and algebraic structures to solve theoretical or real-world problems. Abstract math problems might include finding the area of a plot of land during a <a href='https://www.masterclass.com/articles/land-survey-guide' target='_self' class='mc-text--link' data-mc-segment-trigger='true' data-mc-segment-event-type='Article Page Click' data-event-obj='{"textTag":"p","href":"https://www.masterclass.com/articles/land-survey-guide","linkCopy":"land survey","category":null}'>land survey</a>, solving for the mean <a href='https://www.masterclass.com/articles/cost-of-goods-sold-explained' target='_self' class='mc-text--link' data-mc-segment-trigger='true' data-mc-segment-event-type='Article Page Click' data-event-obj='{"textTag":"p","href":"https://www.masterclass.com/articles/cost-of-goods-sold-explained","linkCopy":"price of goods","category":null}'>price of goods</a> to match consumer spending habits, or visualizing symmetric lines for roadways as a part of <a href='https://www.masterclass.com/articles/urban-planner-guide' target='_self' class='mc-text--link' data-mc-segment-trigger='true' data-mc-segment-event-type='Article Page Click' data-event-obj='{"textTag":"p","href":"https://www.masterclass.com/articles/urban-planner-guide","linkCopy":"urban planning","category":null}'>urban planning</a>.Australian mathematician <a href='https://www.masterclass.com/classes/terence-tao-teaches-mathematical-thinking' target='_self' class='mc-text--link' data-mc-segment-trigger='true' data-mc-segment-event-type='Article Page Click' data-event-obj='{"textTag":"p","href":"https://www.masterclass.com/classes/terence-tao-teaches-mathematical-thinking","linkCopy":"Terence Tao","category":null}'>Terence Tao</a> says, “There is a perception that mathematics is some sort of sorcery.” Abstract math, however, is a highly creative tool that can unlock new modes of thought. “It&#39;s a way of thinking,” he says. In collaboration with mathematician Ben Green, Terence used creative thinking to <a href='https://www.masterclass.com/articles/mathematical-problem-solving' target='_self' class='mc-text--link' data-mc-segment-trigger='true' data-mc-segment-event-type='Article Page Click' data-event-obj='{"textTag":"p","href":"https://www.masterclass.com/articles/mathematical-problem-solving","linkCopy":"solve a problem","category":null}'>solve a problem</a> related to the Twin Prime Conjecture. The pair proved it’s possible to always find a progression of prime numbers of any length and of equal spacing in the infinity of integers.

Abstract Math Examples and Applications

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Abstract mathematics concerns theorems, algorithms, and other mathematical concepts and broad applications for practical use in the real world. Learn about the history of abstract mathematics.

Abstract mathematics concerns theorems, algorithms, and other mathematical concepts and broad applications for practical use in the real world. Learn about the history of abstract mathematics.
