Mathematics: its Development and Abstract Evolution

Ask ChatGPT
The most basic math is 1+1=2. This equation is the foundational cornerstone of Math and it is unassailable. But in math's progression, uncertainties and predictions abound. How could that happen when math is precise? In what situations can there be any uncertainty? How did Math math evolve from "1+1=2" to abstract Mathematical mathematical equations that only the most brilliant minds can decipher?

a Math maze: from 1+1=2, to this!

Progressive Levels of Math

1. Arithmetic - Addition & Subtraction, Multiplication & Division, Place Value & Number Systems, Fractions & Decimals
2. Algebra - Linear Equations & Graphing, Polynomials & Factoring, Quadratic Equations, Exponents & Radicals, Functions
3. Geometry - Architects and engineers use geometry to design buildings, bridges, and other structures. Using triangles, circles, polygons, and the Pythagorean theorem, they calculate angles, areas, and volumes to ensure structural integrity and aesthetic appeal.
   
   Trigonometry - study of triangles and their relationships. Surveyors use trigonometry to calculate distances and angles in land measurement, construction, and mapmaking. Sailors and pilots use trigonometry to determine their position and course. Global Positioning Systems rely on trigonometric calculations to pinpoint your location.
4. Calculus - Calculus, measures the rates of change and accumulation, used to calculate velocity, acceleration, and forces. Engineers use calculus to model the flow of liquids and gases, essential for designing pipelines, airplanes, and ships. Calculus is used to model interest rates, stock market trends, and other financial phenomena. Epidemiologists use calculus to model the spread of diseases. Calculus is used in many machine learning algorithms, particularly in optimization problems.
5. Specialized Mathematics - Beyond these core areas, there are many other specialized branches of mathematics, such as number theory, topology, differential geometry, probability, and statistics. These are often studied at the undergraduate and graduate levels.

Math from Precise to Abstract
Question:
At its foundation, Mathematics is supposed to be precise and unassailable: 1 + 1 = 2, Yet as math progresses into more advanced areas, it becomes abstract, producing uncertainties, approximations, and even paradoxes. How is that possible?

Answer:
From simple arithmetic (counting sheep), Math has evolved from counting numbers to a rich and evolving landscape of ideas full of paradoxes and infinities.

1. The Infinite and the Paradox - Concepts like infinity and infinitesimals (infinitely small quantities) introduce complexities that don't exist in finite arithmetic. Consider the infinite series: 1/2 + 1/4 + 1/8 + 1/16 + ...Each term gets smaller and smaller, approaching zero. The sum of this infinite series is exactly 1. This concept of approaching a value (1 in this case) through an infinite process is a limit, a core concept of calculus.
   
   This also brings up paradoxes like Zeno's paradox, where you can divide a distance into infinite halfs, and it seems like you would never reach the end. To get from point A to point B, you must first cover half that distance, point C. Having done so, you know that from point C to point B, you must first have to cover half that distance. This goes on to infinity, you keep getting closer to point B but you never do. The paradox lies in the fact that in reality, you can cover point A to B. This is where Math does not represent the real world. To mathematically resolve this paradox, you assign the value to this infinite equation, which is 1.
2. Development of Non-Euclidean Geometries - For centuries, Euclidean geometry was considered the only "true" geometry. However in the 19th century, mathematicians discovered non-Euclidean geometries, which challenged fundamental assumptions about space and parallel lines. Multiple Geometries showed that different geometric systems could be logically consistent, leading to a broader understanding of mathematical structures. Euclidean Geometry assumes parallel lines never intersect. But Non-Euclidean Geometry (Hyperbolic) in hyperbolic geometry, parallel lines diverge. This geometry is used to model curved spaces. In Non-Euclidean Geometry (Elliptic Geometry), parallel lines eventually intersect. This geometry is used to model the surface of a sphere. These geometries (Euclidean and Non-Euclidean) show that the "rules" of space aren't absolute but depend on the underlying assumptions (if space is curved or flat or shaped otherwise).

in a hyperboloid, 2 parallel lines diverge, proving a non-Euclidian geometry to be its own sound, logical and viable system


3. Rise of Set Theory and Paradoxes - Set theory was developed to provide a rigorous foundation for all of mathematics. However, set theory itself led to paradoxes like Russell's paradox, which revealed inconsistencies in the foundations of mathematics. Consider the set of all sets that do not contain themselves. Does this set contain itself? If it does, it violates its own definition. If it doesn't, it must contain itself. This paradox revealed a fundamental problem with the naive understanding of sets.
4. Godel's Incompleteness Theorems - In the 20th century, Kurt Godel's incompleteness theorems proved that any sufficiently complex mathematical system will contain statements that are true but cannot be proven within the system itself. Undecidable, meaning neither the statement nor its negation can be proven. This means there are inherent limitations to what can be known and proven within any formal mathematical system.
5. Nature of Mathematical Truth - Mathematics is built on a foundation of axioms (fundamental assumptions) and logical rules. By changing the axioms, mathematicians can create different mathematical systems, each with its own set of truths. Euclidean Geometry is based on axioms like "a straight line can be drawn between any two points." Non-Euclidean Geometry is based on different axioms about parallel lines. Both systems are logically consistent, but they describe different "realities." This means that mathematical "truth" is relative to the chosen axioms.

Math's Foundational Axioms

Question:
Mathematics is built on a foundation of axioms (fundamental assumptions). Can you give examples of these axioms?

Answer:
Axioms are the bedrock of mathematical systems. They are statements that are accepted as true without proof, serving as the starting point for logical deductions and the construction of theorems. These axioms are true within their specific mathematical system. The choice of axioms is often based on intuition, experience, or the observable reality. Axioms are not proven, but they are the basis for all proofs within the system. Here are some examples:

1. There can only be one line passing through 2 points.
2. Any line segment can be extended indefinitely in a straight line.
3. Given any point and any distance, a circle can be drawn with the point as its center and the distance as its radius.
4. All right angles are congruent.
5. Two sets are equal if and only if they have the same elements.
6. For any set of sets, there exists a set that contains all the elements of those sets.
7. 0 is a natural number.
8. For every natural number n, there exists a unique natural number S(n), called the successor of n.

Ending Thoughts
Math people see the world in a very different way, much like an enlightened person sees reality beyond human perception. We don't see that. We can only ponder. Edward Witten is perhaps the world's most brilliant mind now. I don't think he has stepped on solid ground in decades. His entire world is in the ether, pursuing a complex theory that can only be predicted by Math. He is the only non-Mathematician who won the highest award for Math (essentially out-mathing the most brilliant mathematicians). His field of expertise? Theoretical Physics which again does not exist in physical reality. How does he pursue this discipline? Again, through Math. He can explain reality in numbers. What's it like to live like Edward Witten? I would love to be in his shoes for 24 hours.

Another math hero I have is an obscure self-taught Indian guy who is probably the most brilliant mathematician who ever lived - Srinivasa Ramanujan. He never got the due credit he deserved. Why? Because he was Indian at a time when India was a colony of Britain. No British could possibly allow any Indian guy to be smarter than them. I think the Indian people are one of the smartest people on the planet - as smart as the Chinese and the Jews. Now that they have landed a spacecraft on the moon, their national pride is on a momentum. This can only unleash the genius that has long been repressed within them.

I've been blessed with many gifts. But if it's not too much to ask, I would love Math wizardry to be an addition.

--- Gigit (TheLoneRider)
YOGA by Gigit | Learn English | Travel like a Nomad | Donation Bank

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