# Old Math, New Math

Mathematics leads in many directions and to many seemingly strange ideas. If the mathematics behind discoveries is consistent, the ideas are usually portals to new mathematical breakthroughs. Along the way, there have always been mathematicians who would not accept or even consider the possibilities and usefulness of new mathematical ideas that popped out of the expansion of mathematical concepts. Among these are:

• The Pythagoreans refused to consider $\sqrt 2$ and other irrationals as numbers.
• Early Europeans (even during the medieval era and the beginning of the Rennaisance era) generally dismissed the use of Hindu-Arabic numerals in favor of Roman numerals.  They especially didn’t want to consider “zero” as a number — calling the notion of “nothingness as a number” as ludicrous. The only groups of people who use Hindu-Arabic numerals at the time in Europe were bankers and traders. They recognized the advatages of Hundu-Arabic numerals over Roman numerals. Imagine doing long multiplications and divisions using Roman numerals.
• It was Leonardo of Pisa, also known as Fibonacci, who popularized the use of Hindu-Arabic numerals during the 13th Century. In his book Liber Abaci (1202), he argued that the Hindu-Arabic numeral is more superior and flexible than the Roman numerals. Fibonacci was the first person who tried to convince the general public to switch to Hindu-Arabic numerals, and his efforts had paid off. But even then, many people still refused to switch to Hindu-Arabic numerals and several schools refused to include it in their curriculum.
• Most mathematicians of the 16th & 17th century refused to accept negative numbers, let alone imaginary, complex. quaternions, transcendental and transfinite numbers. During this era we have Nicholas Chuquet and Michael Stifel referring to negative numbers as absurd, and Girolamo Cardano giving negative numbers as solutions to equations even though he considered them as impossible answers. Blaise Pascal was reputed to have said “I have known those who could not understand that to take four from zero there remains zero.”, On the other hand. Albert Girard recognized complex numbers as formal solutions to equations which had no other solutions.
• In 1585, mathematician Simon Stevin published a pamphlet titled De Theinde (“the tenth”). The pamphlet explained the value and applications of decimal fractions like for extracting square roots. A number of ideas in the pamphlet was ahead of Stevin’s time. The metric system was introduced more than two centuries after the publication of the pamphlet, but there was already a discussion of decimalization of system of weights and measures in the pamphlet. Stevins stated that it was just a matter of time before the universal introduction of decimal weights, measures and coinage.
• From the late 1860s to the early 1900s, traditional mathematicians refused to accept what they called mathematical anomalies/monsters that surfaced from the works of such mathematicians as Cantor, Wierstrass, Peano, Koch and Sierpinski, which led to today’s evolution of fractal geometry.

### Posted by Edmark M. Law

My name Edmark M. Law. I work as a freelance writer, mainly writing about science and mathematics. I am an ardent hobbyist. I like to read, solve puzzles, play chess, make origami and play basketball. In addition, I dabble in magic, particularly card magic and other sleight-of-hand type magic. I live in Hong Kong. You can find me on Twitter and Facebook. My email is edmarklaw@learnfunfacts.com

## 22 thoughts on “Old Math, New Math”

1. Dear Sir,
I have done BA in Actuarial Science. I have also done Graduate work in Statistics at Columbia University.
I have what I believe is a New Calculus. As it stands, Calculus is built upon incorrect foundations. Moreover, I also have a criticism of the Standard Deviation. I think it is incorrect for all its purposes. So, how do we get what seem like correct results a lot of the times?
I have discussed this in the Philosophy of Mathematics, which is central to the understanding of the flaws and foundations of the current Mathematics.
I have found a new way to find a derivative. I think it is correct that Calculus moves in this direction. Please read the post completely:
On the New Calculus
Some theory (below) before the definitions. This is to convey how I conceptualized the beginning of what I believe is a “New Calculus”. I thought about how an arc of a circle can be defined in a topological sense. This led me to the definition of a line, and eventually, a circle, square, rectangle, and equilateral triangle. Therefore, I want to convey the initial part, as it is paramount to the understanding of what follows in terms of definitions.
A circle is formed by very short lines ( not horizontal or vertical, but “angulated”) on the circumference, as the limit of those lines, “delta x,” go to zero. So, the circumference of a circle is the sum of these small lines that tend to go to zero in the limit. Importantly, though, these lines are not horizontal or vertical, thereby not providing epsilon as the limit (as they go to zero in the limit).
Suppose that there are two points on a circle’s circumference. The curvature between the points comprises of small lines “delta x”, where the limit of each delta x goes to zero.
If the distance between the curvature is formed by “angulated” distances (very small), then the curvature (or the arc would never form). IT would be in a sense chiseled shaped object. As delta x goes to zero (x being the distance between two points on the circumference of a circle), the sum of these lines forms the circumference of a circle.
So, the circumference of a circle is a sum of pieces (or distances) X, where the limit of delta X goes to 0.
It is summation of delta x where the limit of delta x goes to zero. Or X the curvature is the sum of pieces delta x as the limit of each delta x goes to zero
The summation of delta x as x goes to zero is the circumference of the circle
From a point X on the circumference of a circle, the distance between point X and point Y (on the circumference) is the summation of delta x’s as the limit of each delta x goes to zero.
A straight line (whether vertical or horizontal ) defined upon a certain space is formed of adjoining distances, delta x, as the limit of each distance delta x goes to zero resulting in an error e. The summation of (0 to n
)of these e’s= ne. Therefore, the length of a line over a certain space can be defined as ne
A square can be defined by its diagonal, lets call it s.
This is because all sides are equal and the diagonal determines the length of the sides.
For a square which can be defined by its diagonal:
Sum of delta x as the limit of delta x goes to zero.
(The diagonal)^2= (Summation (0 to n) as the limit of each delta x goes to zero)^2. + (summation (o to n) as the limit of each delta x goes to zero)^2
Thus the diagonal= Square Root( (summation (o to n) of e)^2+ Square Root (Summation (o to n) of e)^2
diagonal= square root( (ne)^2+(ne)^2)
Thus the diagonal = sqrt(2*(n *e)^2)
Thus the diagonal= sqrt(2)*en
A circle’s Circumference= 2 pi r
Circumference= 2 pi ((Summation (0 to n)(lim as delta r goes to zero)
Circumference= 2 pi (Summation (0 to n) of e)
Circumference= 2 pi (ne)
The above definition works because even if r is not a horizontal line, it should give the same result as a horizontal line because all the lines from the center of the circle to the cirumference comprise of the r, and thus should provide an equivalent result. This is a necessary and sufficient truth, that is, by definition it follows that all the “rs” should provide the same result.
Moreover,
definition of (r)=circumference/2pi= definition of a circumference/2pi=2pi(ne))/2pi=ne
definition of an area of a circle:
Area of a circle= pi (r^2)
definition of the area of a circle= pi (summation (n=0 to n=n)(limit of delta x as it goes to zero)^2
=pi(en)^2
A rectangle’s diagonal can be defined by:
Diagonal ^2= ( Summation (n=0 to n=n)(limit of delta x as it goes to zero)) ^2+(summation (j=o to j=k) (limit of delta y as it goes to zero)^2
Diagonal^2= (Summation from 0 to n of e)^2+(summation from j=0 to j=k of e)^2
Diagonal^2= (ne)^2+ (ke)^2
Diagonal^2= n^2(e^2) + (k^2)e^2
Diagonal^2= e^2(n^2+k^2)
Diagonal= e* sqrt(n^2+k^2)
where n is not equal to k
Equilateral Triangle
Altitude of an Equilateral triangle =(1/2) * √3 * a
where a is the length of the side. Lets call the altitude h
a= 2h/Sqrt(3)
a=2/sqrt(3) *{(sum (o to n))(limit as delta h goes to 0)
a=2/Sqrt(3) *en
perimeter of an equilateral triangle = 3a
definition of perimeter= (6/sqrt(3)) *en
Isoceles Triangle: Let the 2 identical sides be ‘b’. Lets call the height of the triangle as h. Lets call the base (the third side) ‘a’.
Then h= sqrt (b^2-1/4(a^2))
h^2=b^2-(1/4)(a^2)
b^2= h^2+ (1/4)(a^2)
b=Sqrt (h^2+(1/4)(a^2))
definition of b= sqrt((en)^2+1/4(en)^2)
The above follows because a is a horizontal line and h is a vertical line.
definition of b= Sqrt(5/4(en^2))
definition of b= Sqrt(5/4) *en
definition of b= sqrt(5)/2 *en
Also,
definition of b^2= h^2+(1/4(a^2)
=(en)^2+(1/4(en)^2)
=5/4(en)^2
Therefore, definition of b=
b= sqrt(5/4) en
b=sqrt(5)/2*en
definition of h= en
definition of a= en
perimeter= a+2b
definition of the perimeter= definition (of a)+ definition (of 2b)
definition of the perimeter= en+2 ((Sqrt(5)/2)en)
definition of the perimeter= en+sqrt(5)en
surface area of a cube= 6* (l^2) where l is the length of the side
definition of a surface area of the cube= 6* (en)^2
definition of a length of the side of a cube= en
Note: the e, epsilon,is either horizontal or vertical when defining limits.
Special right triangle 45-45-90 with two legs equal inscribed at the origin: (en)^2+(en)^2= hypotenous^2
hypotenous^2=2(en)^2
Hypotenous=Sqrt (2) en
Therefore, the definition of the line y=x is Sqrt(2)en
This is because the line y=x cuts the origin on the cartesian plane in two pieces of 45 degree angles and is represented by the hypotenuse in the above theory.
Similarly y=-x has the same definition. This is because space can exist but the contrary doesn’t hold true. Non-existence is not a finite quantity.
Inscribing a circle with center at origin will not produce definitions of lines, such as y=2x, as a circle has tangible boundaries, whereas the special right triangle (or its diaganol in this instance) on the positive side of the cartesian plane can stretch as long as possible, even infinitely longer. In other words, the line y=x would have bounds within the circumference of a circle. It wouldn’t define it appropriately.
Among the transformations, translation doesn’t change the definition of an object. Therefore, where the shape is placed on the Cartesian please doesn’t change its definition.
This is easily understood with the basic example:
derivative of y=nx+a
dy/dx= n
Therefore, the position of the object along translations doesn’t change the derivative (whether it is a positions sideways or b positions high or low). Likewise, for the definition of the derivative y=nx+a would not be different since y=x+a has the same definition as y=x. Therefore, the translations of objects wouldn’t change their definitions.
Furthermore, lets equate one unit of the line with one unit on the Cartesian Plane. Then, a line defined on the plane becomes intuitively more accessible.
For instance, the definition of a horizontal line defined upon a space (0 to 10 on the Cartesian Plane for the sake of assumption) would be 10e (that is, n=10).
This is not a necessary and sufficient truth as we can also define 2 units on the Cartesian plane and equate it with each of n pieces of a line.
However, equating one unit on the Cartesian Plane with one e on the line seems to be a reasonable template to further develop these ideas.
Once we accept the assumption that one unit on the cartesian plane corresponds with one unit of the line en, then half a unit of the line (or, say, 9.5) will correspond with .5e. This is because the piece e is proportional to its distance on a piece of a line (this is because the line is either vertical or horizontal in this case).
N.B Definition of A = def (B+C)
Def A=def (B) +def (C)
Similarly,
Definition of A= def (B-C)
Def (A)= Def (B)- Def(C)
Def(A)= Def (B+C)
Def(A)/a=Def (B+C)/a where a is any positive number
Def f deravitave of x= (def(f(x)+ delta x) -def (f(x))/ def (delta x)
Where limit of delta x goes to zero.
Suppose f(x)=x
Definition of f(x)= lim as delta x goes to zero (def (f(X)+delta x))- def f(x))/def (delta x)
def (f(X))= srt(2)en
def (delta x)=e
Def(f(x)+delta x)= Sqrt(2) en+e
DEf (f(x)+delta x)-def (f(x))= Sqrt(2)en+e-Sqrt(2)(en)
DEf (f(x)+delta x)-def (f(x))/def(deltaX)= (Sqrt(2)en+e-Sqrt(2)(en))/e
=e((Sqrt(2n)+1-Sqrt(2n))/e
=1
Note that when we subsitute Sqrt(2) en as the definition, we have already taken the limit to zero of f(x) or f(x+delta x)
for the definition of the deravative of y=2x
def 2(x+deltax)-def (2x)/def (delta x)
2 def(x+delta x)- def (2x)/def(delta x)
I have mentioned in the above theory how to find the definition of the line y=2x.
Since
Def (A)=Def(aB) where a is an integer
Then Def(A)= a(Def B)
Therefore it follows,
def y=def (2x)
def y=2( Def x)
def y= 2*Sqrt(2) en
=2^(3/2) en
def (f(X))=2^(3/2) en
Def(f(x)+delta x)= 2(f(X)+delta x)= 2(Sqrt(2) en+e)
Def (delta x)= e
Therefore, definition of a derivative= (2(Sqrt(2)en+e))-2^(3/2)en)/e
= ((2^3/2)en+2e_ -2^(3/2)en)/e
=2e/e=2
For the definition of the derivative of y=nx
definition of nx= n Sqrt(2) en
DEf(f(x))= definition of nx= n Sqrt(2)e(n1)
where n1 is the subscript to distinguish from the other n
Def (F(X+Delta X)= n(Sqrt 2*e(n1)+e)
Def (Delta X)=e
Therefore, the definition of the derivative is
n((Sqrt(2)*e(n1)+e)-n Sqrt(2)*e(n1)/e
n(Sqrt(2)* e (n1)+e- Sqrt (2)*e(n1)/e
Then n(e)/e=n
Therefore, the definition of the drivative nx= n
N.B The limits of delta x approaching zero are embedded in the definitions.
On the Philosophy of Mathematics:
Please further note that the “New Calculus” shows that the laws of Mathematics are universal, that despite the differences in this calculus, we get equations which are workable and mean something. These laws of Mathematics exist in nature, just like the laws of physics. We decode these laws based upon logic,too, but the very birth of these ideas is mostly intuitive. Even then, so, we “discover” abstract Mathematics which exists in space, truthful, but we can only grasp this through our basic intuitions at first, and “then” justify it through logic. But something should not emerge correct based on something that is incorrect, however, since these are the laws of Mathematics (which in its truest form exist in nature), they can be approached somewhat correctly from wrong foundations. This is the proof for the very existence of these laws, that despite the flaws in the Old Calculus, we could get correct results a lot of the times. This is because we accessed that part of Mathematics in nature that we could have approached from those incorrect foundations. If there weren’t these laws, then the Old Calculus would never have come to be and all the Mathematical foundations would have to be 100 percent correct for Mathematics to work (even) slightly.
For example def derivative= def (ax)= a def(x) where a is an integer.
However, how do we come to accept this assumption?
We do trial and error with equations and realize that this works. Then we accept this assumption based upon the evidence that it leads to solid conclusions. This is based on intuition and discovery rather than being a self-evident truth. Thus, we discover Mathematics rather than create it ourselves. And, since we discover Mathematics, it exists with its laws in nature just like the laws of physics.
For instance, E (aX+b)=aE(x)+b follows from integrands which have an axiom that integral of (a* f(x)) with limits from b to c is the equivalent to (a*(integral of f(x) with limits from b to c)). This is one of the foundations of calculus, but the rule has its justification in discovery rather than it being self-evident (that is, even if it is axiomatic, it is not self-evident).
If we couldn’t see 2+2=4 in this universe, then it would be impossible to know what 2+2 would amount to; the only way to know it is 4 in this universe is to add two objects with two other objects. Once, then, we know it is 4, we can go further and claim 1 and 3 equal 4 using the logic: that is, half of two is one and 1.5 of 2 is 3. Therefore, 1 and 3 add up to 4. Logic follows from experience. In this case the discovery is that someone saw 2 objects and 2 other objects, and realized 2 times 2 is 4. Then we could realize what 4 amounts to. And, then we realized that 3 is a so and so fraction of 4, and 1 is so and so fraction of 2 (or 4).
In other words these numbers exist in nature and are (were) a matter of discovery rather than being based on the notion that their properties are self-evident.
We can imagine a universe without any objects; in this universe there wouldn’t be any laws of Mathematics since the concept of a number wouldn’t exist. There would be no logic. But since the universe exists with objects in space, there is logic and Mathematics. Mathematics is conditional on the existence of something, a finite quantity.
In other words there will not be the notion of divisibility in a universe with no objects. Since objects exist in space, there is Mathematics. Thus, the existence of Mathematics is conditional on the existence of the finite–and “hence” limit of delta x going to zero is something finite and not the non-existence of space, that is zero– and thus, discovery. It is a discovery because there are multiple objects in space from which we infer that, say, 2+2=4
Laws of mathematics vanish when we consider a universe without an object, and as the objects in the universe are created (discovered), so are the laws of Mathematics.
e(n+5)- e(n)=5e
e(n+a)-e(n)= a*e
This is because there are a spaces between e(n+a) and en, with each of those spaces going in the limit to zero and yielding e.
For the definition of derivative of x^2= def f(x+delta x)- def (f(x))/e
therefore the definition of ((x+deltax)^2-def (x^2))/e
Therefore, (def (x^2)+ def(2*x*delta x)+def (delata x)^2-def(X^2))/e
Therefore, (def(2*x*delta x)+def (delta x)^2)/e
def (Delta x)^2 is the definition of a parabola.
Like other objects that have been defined before, the definition of the parabola would remain the same regardless of where it is situated on the cartesian plane. So, we can consider the definition of a parabola which is inscribed at the origin, and replace it in the equation above.
(x-h)^2 =4p(y-k)
focus is (h,k+p)
directrix y=k-p
vertex is (0,0)
(defy=en)
def (k-p)=en
focus on (0,0)
k+p=0
k=-p
x^2=4p(y-k)
x^2=4 p(y+p)
=4py+4p^2
y=0
x^2=4p^2
def (y)= en
y-k=-p
k-y=p
defk-en=defp
x^2=4p(y-k)
x^2=4p(k-p-k)
x^2=4p(-p)
x^2=-4p^2
def (x^2)=-def (4p^2)
Since p is a point on a line, the definition should be e
def (X^2)=-4e^2
Therefore, (def(2*x*delta x)+def (delta x)^2)/e
Therefore, ((def(2*x*delta x) -4e^2)/e
Moreover, def (2x*deltax)/e-4e
N.B: When we solve limits that can be evaluated using conjugates, how do we know which result is correct?
For instance,
A typical example is: Evaluate lim as x goes to 4 of ((x^1/2)-2)/(x-4)
Right here when we look at the limits, it comes out as 0/0 (when using the Newtonian Calculus). Then we use the conjugates and the result of the limit is 1/4. However, how do we come to accept the second conclusion and disregard the first conclusion? Because it gives, we think, an answer that is correct. However, from a logical point of view both the conclusions have merit (when thinking using the Newtonian Calculus.
This is why I am making the argument that there are problems with the old calculus and the justifications aren’t terse. It is by trial and error that we realize what works and what doesn’t. What is accepted, then, becomes part of the books. Calculus is based on incorrect foundations and has a lot of logical inconsistency. I encourage people to work at what I am calling the “New Calculus”.
def y=x+a Where a is an integer is def y=x since y=x+a is a translation y=x and the definitions don’t change under translations.
As long as a line is bounded within space, it is a partial line. Definition of say y=x requires the definition of the complete line.
On The Old Calculus:
The derivative of with respect to x is the function and is defined as,
f'(x)= lim (as h goes to zero) ((f(x)+h)- f(x))/h)
The convention is that as the limit of h goes to zero, it implies that h=0. However, since it is the limit that goes to zero, it should provide us with some finite quantity, or existence of space, which can be called e (epsilon).
When we take derivatives defined upon a certain space, we disregard the error, epsilon. Where there is epsilon in the denominator (of a derivative function), and we assume it is 0, then the derivative doesn’t exist.
Since the error is minute in the denominator, the numerator does go to (almost) infinity (where it works), thereby providing a similar result as the assumption that as h goes to zero in the limit, the result is infinity (or the limit of the derivative doesn’t exist). However, it is not technically correct to think that way.
IF we find a derivative of a function, where the denominator, h, cancels out with some quantity in the numerator (and results in a function without h), then the epsilons also cancels out, thereby providing the same results as the derivative of the function.
The third possibility is where h, the denominator, cancels out with some quantity in the numerator, resulting in a function whose limit goes to zero (of h)where both h and x comprise of the function. In this case, since the limit of h goes to zero, it is assumed that these values become zero (however, in this case, this is incorrect, as lim as h goes to zero, it results in the error e, which remains part of the derivative).
If the derivative has some function with epsilon (say X+e), then the integral would give the function X^2/2+ ex+c, thereby making a lot of calculus incorrect. However, c in this would absorb ex, thereby providing the illusion that the integral is correct. When we take limits of integrals, though, there are discrepancies. If the integral is Xe,and suppose we take the limit from 1 to 5,. then the resulting function is 5e-e=4e.
Lets think of h in the correct manner which is delta x.
The limit of h (or delta x) in the derivative as it goes to zero is e because delta x (or change in x) is one piece of a line that goes to zero in the limit.
The difference between a line (defined upon a certain space) is that it is made up of many pieces (n pieces) of delta x as each delta x goes to zero in the limit thus providing the definition en.
On the Standard Deviation:
There are major flaws in the usage of the Standard Deviation in either measuring consistency in an event or measuring the risk in assets.
When measuring consistency there should be a marker above which the data is the marker. This marker could be the average of the data.
For instance, if a the average score is 50, and there are two scenarios: one batsman has scores of 50,50,50,50 and the other one is 50, 50, 50, 400; count 400 (being above 50) as 50 to get the desired result in measuring how consistent the player is.
The second player (in the above) would be regarded as more inconsistent using the Standard Deviation, whereas heu is intuitively clearly more consistent (using the word consistent in the conventional sense). The correct way to think is that the second player is at least as consistent as the first player, but the SD would give incorrect results.
By conventional logic someone scoring 25 each in 5 innings would be more consistent than someone scoring 49, 50, 50, 50, 50.
Another Example:
Standard Deviation is widely used to measure risk in assets. I think a higher standard deviation is considered a risky bet, with greater upside and downside potential.
However, if the asset A’s returns are: 50, 50, 50, 400 and Asset B’s return are 50, 50, 50, 50– the latter will have much less standard deviation and risk (in fact zero) while the former (Asset A) will have a much higher standard deviation and ‘be full of risk’. But we can clearly see that Asset A’s downside potential is a lot less that Asset B’s, and it is a safer, or less riskier investment than Asset B.
These are a few data points so we can intuitively see which bucket has more risk, but what happens if such a scenario is happening over 100s, 1000s of data points?
In other words the Standard Deviation is subtracting the points above the mean from the mean and lower than the mean from the mean and after squaring and adding the differences Calculating risk. However, a point below the mean may imply lesser risk as a point higher than the mean might imply a higher risk, however adding these points after subtracting from the mean and then squaring them would be a great distortion, that is, the volatility would jump through the roof. In other words points that should reduce risk (by being very low in the supposed instance) exacerbate risk as the differences between the mean and the point are accentuated and thereafter squared.
Since the differences between the data and the mean are squared, negative values are squared to produce results which are positive (it becomes additive to the values higher than the mean that are also squared and then added). And, if the points have a larger Deviation from the mean (downside),squaring them mean further amplifying and distorting the results as these differences become greater when squared.
Think of a set of Betas ranging from low to high. In this example lower Betas Represent lower risk and higher Betas represent more risk. When we calculate the SD of the Betas, the points below the mean (the safer ones) are actually causing the Standard Deviation to increase and the lesser the beta, the more pronounced the effect or risky it becomes when subtracted from the mean and squared. Thus, lower betas are increasing the risk, the lower they are from the mean, the more the risk.
If the beta paradigm is incorrect, then, by assumption, SD is incorrect. If it is correct, then still the SD is incorrect.
This implies that the SD is incorrect.
On The Philosophy of Mathematics:
Please further note that the “New Calculus” shows that the laws of Mathematics are universal, that despite the differences in this calculus, we get equations which are workable and mean something. These laws of Mathematics exist in nature, just like the laws of physics. We decode these laws based upon logic,too, but the very birth of these ideas is mostly intuitive. Even then, so, we “discover” abstract Mathematics which exists in space, truthful, but we can only grasp this through our basic intuitions at first, and “then” justify it through logic. But something should not emerge correct based on something that is incorrect; however, since these are the laws of Mathematics (which in its truest form exist in nature), they can be approached somewhat correctly from wrong foundations. This is the proof for the very existence of these laws, that despite the flaws in the Old Calculus, we could get correct results a lot of the times. This is because we accessed that part of Mathematics in nature that we could have approached from those incorrect foundations (where it works). If there weren’t these laws, then the Old Calculus would never have come to be and all the Mathematical foundations would have to be 100 percent correct for Mathematics to work (even) slightly.
For example def derivative= def (ax)= a def(x) where a is an integer.
However, how do we come to accept this assumption?
We do trial and error with equations and realize that this works. Then we accept this assumption based upon the evidence that it leads to solid conclusions. This is based on intuition and discovery rather than being a self-evident truth. Thus, we discover Mathematics rather than create it ourselves. And, since we discover Mathematics, it exists with its laws in nature just like the laws of physics.
For instance, E (aX+b)=aE(x)+b follows from integrands which have an axiom that integral of (a* f(x)) with limits from b to c is the equivalent to (a*(integral of f(x) with limits from b to c)). This is one of the foundations of calculus, but the rule has its justification in discovery rather than it being self-evident (that is, even if it is axiomatic, it is not self-evident).
If we couldn’t see 2+2=4 in this universe, then it would be impossible to know what 2+2 would amount to; the only way to know it is 4 in this universe is to add two objects with two other objects. Once, then, we know it is 4, we can go further and claim 1 and 3 equal 4 using the logic: that is, half of two is one and 1.5 of 2 is 3. Therefore, 1 and 3 add up to 4. Logic follows from experience. In this case the discovery is that someone saw 2 objects and 2 other objects, and realized 2 times 2 is 4. Then we could realize what 4 amounts to. And, then we realized that 3 is a so and so fraction of 4, and 1 is so and so fraction of 2 (or 4).
In other words these numbers exist in nature and are (were) a matter of discovery rather than being based on the notion that their properties are self-evident.
We can imagine a universe without any objects; in this universe there wouldn’t be any laws of Mathematics since the concept of a number wouldn’t exist. There would be no logic. But since the universe exists with objects in space, there is logic and Mathematics. Mathematics is conditional on the existence of something, a finite quantity.
In other words there will not be the notion of divisibility in a universe with no objects. Since objects exist in space, there is Mathematics. Thus, the existence of Mathematics is predicated on the existence of the finite–and “hence” limit of delta x going to zero is something finite and not the non-existence of space, that is, zero– and thus, discovery. It is a discovery because there are multiple objects in space from which we infer that, say, 2+2=4
Laws of mathematics vanish when we consider a universe without an object, and as the objects in the universe are created (discovered), so are the laws of Mathematics.
Because Mathematics follows from discovery, I am certain that the new calculus will flow. If the wrong foundations led us to right conclusions a lot of the time, then the right Calculus will come out with equations that mean something, too. We just have to discover Mathematics again.
The difference between the laws of physics and the laws of Mathematics is that the latter are universal whereas the laws of physics can change with the nature of substances.
On Probability Theory:
If the probability of an event on a continuous probability density function is 0, then all the possible collection of events have zero probability, too (think of infinite pieces each with probability zero), which is a contradiction as the probabilities of events should add to 1.
How can an event, which has zero probability, happen?
Probability that x=h is not zero on the curve— it is epsilon, where epsilon is less than or equal to 1 and greater than or equal to zero.
It is a contradiction that the probability of an event A on a curve is zero yet its existent is a possibility.
The function is continuous while the probability of an event (X=a) is not zero, but epsilon, where e is greater and equal to zero and less than or equal to 1.
Sum of epsilons is 1.
Suppose that there is a countinous, probability density function f(x). The probability that X takes a certain value, lets say A, is not zero. There is existence of space, a finite quantity, epsilon (unless epsilon is zero).
The sum of e’s from zero to infinity is 1. The summation of e’s goes to infinity because there are infinite number of events. The e’s don’t have the same value, therefore, it is the summation of e’s to infinity where every e has a subscript consistent with the value of the summation. So, it is summation of en (where n is the subscript) as the summation goes to infinity.
If the probability of an event P(X=a) where a is a finite number on a continuous probability density function is 0 (if it follows by definition or assumption, not a fact), then all the possible collection of events have zero probability, too (think of infinite pieces each with probability zero), which is a contradiction as the probabilities of events should add to 1. In other words the assumption that an event A by “necessity” has zero probability on a probability density function is incorrect.
The sum of en’s (where n is the subscript) from zero to infinity= 1
Lim n goes to infinity as summation goes to n of en (where n is the subscript of e)=1
This doesn’t imply that a probability density function has a probability at P(X=a) which is the length of the vertical distance from a to the curve divided by the area of the region under the curve. This is because there are infinitely many points, and thus probabilities, which would add to more than 1
Also,
Limit as P(X=a) goes to zero is e+
Limit as P(X=b) goes to zero is e1
+….+
Limit as P(X=n)= en…
+ (Limit as P(X=infinity)= e (infinity)= 1
Continuous density function can be divided in infinite number of pieces.

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2. Intriguing read. I wish you to post a blog that discusses the ‘use of mathematics in medicine and biology’.

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3. Ooutstanding post on the many exciting facets of mathematics! I have been following your amazing post for quite some time now, but never received any notifications. But now the problem has been taken care of by WordPress.

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1. Thanks for reading. I am having problems like that as well. And the reader doesn’t show all the blogs that I follow for some reason…

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4. We have been – still are — so quick to say “nay”…

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5. Pie is also irrational right? so what about the approximation 22/7 (ah this has always confused me)

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1. Pi is both an irrational number and a transcendental number (there’s a difference).

22/7 is equal to 3.14285731428573142857… Note how the decimals repeat indefinitely after every six decimals. This is called non-terminating recurring decimal. So, it’s a rational number.

A more simplistic way of looking at it is this: If it can’t be accurately represented with a fraction, like pi, then it’s irrational, if it can, like 22/7, then it’s rational.

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1. Yes I got it. Very interesting :D
Thank you for the enlightenment, have a great day :)

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2. You are right its irrational and transcendental.
People in many countries used different way to represent pie as an rational number which is easy for calculation. Most believe π is equal to 22 over 7. Just an representation. Many rational serves that.
π = 3.141592653589793….
Best rational representation is fourth root of 2124 over 22 which is equal to 3.141592653.
(Used in India during 1900s)
Thank You Math world.
Thank You Remark M. Law

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6. An enjoyable read. Note that Roman Numerals persist to this day as the copyright year for old films…use it to maintain my useless roman numeral reading skills.

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1. Indeed. Also, you can still see them on some analog clocks, book chapter numberings, outlines in complex documents, etc.

And I have a friend who’s the sixth gen. in his family who bear the same first name. So, there’s a “VI” at the end of his name :)

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7. I love it! My blog post today is about education and I was literally typing the word Maths when this popped up in my email! ☺ Great stuff x

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