Isn't the definition of infinitesimal contradictory?

[wtf]

So we are in the same page.

I think so. SOME math is "unreasonably effective" in the physical sciences, in the famous words of Eugene Wigner. And SOME math is a complete fiction, having no possible relation to the world.

How can we tell these two apart? Very often an area of math that has no possible relation to the world suddenly becomes relevant. A striking example is number theory and the study of integer factorization. This was first studied over two thousand years ago and for all that time was regarded as utterly useless, although of great theoretical interest. Then suddenly in the 1980's, computers became an integral part of society and somebody invented public key cryptography, which is based on the theory of factorization. Something regarded as completely useless for two thousand years suddenly became a vital part of society. It's amazing when you think about it.

By the way, what is your opinion about infinitesimal? Our discussion was disrupted. I still don't understand how you could define a lower limit for infinitesimal on real number domain when the limit is zero.

I don't understand what you mean. There IS NO LOWER LIMIT for an infinitesimal. If you have a system i which there are infinitesimals, you can always divide an infinitesimal in half to get a smaller one. There is no smallest infinitesimal. Of course 0 is the mathematical limit, but infinitesimals can keep getting closer and closer to zero.

The philosophical issues are a mystery. We don't know why calculus works. Newton and Leibniz thought in terms of infinitesimals, but they had no logical explanation to make infinitesimals logically correct.

In the following 200 years, mathematicians finessed the subject by inventing the theory of limits, which replaces the idea of "infinitely close" with the more workable idea of two numbers being "arbitrarily close." So infinitesimals were banished from math.

In the 1960's, the theory of infinitesimals was rescued and made logically correct. But as I noted, we still teach calculus and do advanced math using the traditional theory of limits. Infinitesimals have not gained much if any mindshare.

This would all change if tomorrow morning professor so-and-so in Helsinki proves that P = NP using infinitesimals. Then everyone would become interested in infinitesimals.

Until this happens, limits are in and infinitesimals are out. But these things are historically contingent. There's no telling how future mathematicians will think about things.

As far as the philosophical issues, there's a mystery. It goes back to Euclid. Euclid said that a line is made up of points. But points are dimensionless and have no size; yet lines have both dimension and size.

How do you put together an infinite bunch of dimensionless points of size zero to get a one-dimensional line of any size you like? And for that matter, we can put together infinitely many points to make solid objects of arbitrary volume in n-dimensional space for any value of n.

It's a puzzler. One that no mathematical theory can answer. And it relates to the nature of the world, the question of whether the universe is ultimately continuous or discrete. Nobody knows.

[-1-]

mathematical formalism fails, because it says that ALL mathematics is nothing more than symbol manipulation. That fails to explain the power of mathematics in describing the world.

Math formalism does not fail as long as it stays clear of trying to describe reality.

To describe reality mathematics is useful, when you think of it the way our grade 9 teacher introduced the concept back in my highschool days in Hungary. Erzsi Neni (Elder Miss Elizabeth) said, "quantities consist of a unit, and a number denoting the units."

One apple, two oranges, square root of two centimetres, pi metres, sixteen billion light years, etc.

The real world can be described with the aid of numbers, and then mathematicians who specialize in this field, take the math of some science, strip it of its attachment to reality, manipulate the numbers on their own, in a fashion of mathematical formalism; when they had enough of it, they reattach the units to the numbers, and claim that the resultant is reality.

This is how Quantum Mechanics was born.

I still reject the impossibility that QM is a huge hoax. In other words, I entertain the idea, that QM is an interpretive science with miraculously perfect predictive capabilities, yet its concepts are ill-begotten because the math was manipulated in a formalistic way, and I feel, but can't prove, that coming back to reality and reapplying the math results but now with units is actually a very wrong way to go about it.

[wtf]

mathematical formalism fails, because it says that ALL mathematics is nothing more than symbol manipulation. That fails to explain the power of mathematics in describing the world.

Math formalism does not fail as long as it stays clear of trying to describe reality.

Isn't that exactly what I said?

Yet math is "unreasonably effective" in the physical sciences, in Wigner's famous phrase. That's the mystery.

[-1-]

mathematical formalism fails, because it says that ALL mathematics is nothing more than symbol manipulation. That fails to explain the power of mathematics in describing the world.

Math formalism does not fail as long as it stays clear of trying to describe reality.

Isn't that exactly what I said?

Yet math is "unreasonably effective" in the physical sciences, in Wigner's famous phrase. That's the mystery.

I did not deny the validity of you assertion, wtf. I just added the extra dimension of understanding that math is useful if you connect numbers to reality by describing quantities of physical manifests in terms of unit and a number that tells how many units.

I believe the entire discussion has been skirting around the issue of units and numbers of units, but nobody spelled it out before my post. People talked about math, but not how math is only useful in real life science drama when you use numbers as a determinant of the amount of a unit.

It's the first I hear Wigner's Lemma; or even his name. He is a genius.

[-1-]

Sorry. Please strike "He is a genius." and substitute "He or she is a genius."

How sexist of me.

[bahman]

So we are in the same page.

I think so. SOME math is "unreasonably effective" in the physical sciences, in the famous words of Eugene Wigner. And SOME math is a complete fiction, having no possible relation to the world.

How can we tell these two apart? Very often an area of math that has no possible relation to the world suddenly becomes relevant. A striking example is number theory and the study of integer factorization. This was first studied over two thousand years ago and for all that time was regarded as utterly useless, although of great theoretical interest. Then suddenly in the 1980's, computers became an integral part of society and somebody invented public key cryptography, which is based on the theory of factorization. Something regarded as completely useless for two thousand years suddenly became a vital part of society. It's amazing when you think about it.

Interesting.

By the way, what is your opinion about infinitesimal? Our discussion was disrupted. I still don't understand how you could define a lower limit for infinitesimal on real number domain when the limit is zero.

I don't understand what you mean. There IS NO LOWER LIMIT for an infinitesimal. If you have a system i which there are infinitesimals, you can always divide an infinitesimal in half to get a smaller one. There is no smallest infinitesimal. Of course 0 is the mathematical limit, but infinitesimals can keep getting closer and closer to zero.

The philosophical issues are a mystery. We don't know why calculus works. Newton and Leibniz thought in terms of infinitesimals, but they had no logical explanation to make infinitesimals logically correct.

In the following 200 years, mathematicians finessed the subject by inventing the theory of limits, which replaces the idea of "infinitely close" with the more workable idea of two numbers being "arbitrarily close." So infinitesimals were banished from math.

In the 1960's, the theory of infinitesimals was rescued and made logically correct. But as I noted, we still teach calculus and do advanced math using the traditional theory of limits. Infinitesimals have not gained much if any mindshare.

This would all change if tomorrow morning professor so-and-so in Helsinki proves that P = NP using infinitesimals. Then everyone would become interested in infinitesimals.

Until this happens, limits are in and infinitesimals are out. But these things are historically contingent. There's no telling how future mathematicians will think about things.

As far as the philosophical issues, there's a mystery. It goes back to Euclid. Euclid said that a line is made up of points. But points are dimensionless and have no size; yet lines have both dimension and size.

How do you put together an infinite bunch of dimensionless points of size zero to get a one-dimensional line of any size you like? And for that matter, we can put together infinitely many points to make solid objects of arbitrary volume in n-dimensional space for any value of n.

It's a puzzler. One that no mathematical theory can answer. And it relates to the nature of the world, the question of whether the universe is ultimately continuous or discrete. Nobody knows.

So what happen in 1960? How Euclid formulate the problem makes no sense to me.

[wtf]

Interesting.

High praise. Thank you. I make no claim of omniscience but I do try to be interesting.

So what happen in 1960? How Euclid formulate the problem makes no sense to me.

Edwin Hewitt figured out how to make infinitesimals logically rigorous in 1948. Abraham Robinson used Hewitt's idea to develop nonstandard analysis in 1961.

The point about Euclid is that he said a line is made up of points. But the philosophical question is, how can that be? Points are dimensionless and have size zero. Lines have both dimension and length. How do points, no matter how many, become a line?

[bahman]

Interesting.

High praise. Thank you. I make no claim of omniscience but I do try to be interesting.

Interesting.

So what happen in 1960? How Euclid formulate the problem makes no sense to me.

Edwin Hewitt figured out how to make infinitesimals logically rigorous in 1948. Abraham Robinson used Hewitt's idea to develop nonstandard analysis in 1961.

The point about Euclid is that he said a line is made up of points. But the philosophical question is, how can that be? Points are dimensionless and have size zero. Lines have both dimension and length. How do points, no matter how many, become a line?

Thanks. I will read on those topics you introduced.

[wtf]

Thanks. I will read on those topics you introduced.

Don't worry too much if you can't find any useful literature. The mathematical construction of the hyperreals is pretty technical. And the question of how points make up a line is more of a curiosity. I don't think anyone spends much time writing about it. There's a philosopher named Peirce (that's the spelling, not a typo) who had some interesting thoughts on infinitesimals and the continuum.

I found some links on the topic of how points can make up a line.

https://math.stackexchange.com/question ... ional-line

https://www.reddit.com/r/math/comments/ ... s_have_no/

https://www.quora.com/Is-a-line-made-up ... ite-points

[Eodnhoj7]

I'll take a stab, and risk sounding repetitive, but these "phenomenon" as numbers take a role as boundaries which literally give direction to certain movement. For example if I quantify a specific phenomenon, what I am doing is observing a specific set of relations of parts (with the numbers themselves acting as parts) that exist in a localized part of space. This quantity, let's say "4", projects in a linear manner through the timeline until the quantity changes (to 3 or 5 for example) at which case the quantity as "direction" changes. All quantifiable objects maintain at minimum a 1dimensional movement through time and in this manner all quantities exists as directed movement.

Yes that is a stab (in your own unique style) at explaining the nature of the existence of abstractions such as numbers. But a number is a "concrete abstraction," if you will. The number 4 is an abstraction, but everyone has a direct physical experience of the number 4.

All direction is a concrete abstraction in the respect it is founded in "1 dimension". While a number may be a concrete abstraction, when used in the terms of defining physical objects it meres the same change as the physical objects it is defining. So if I count 3 oranges and then count 4, this quantification takes on a direct linear form of 3 tending to 4 considering the act of quantification (applying limits) exists through time as time.

But when it comes to higher mathematical abstractions such as groups, your response gives no insight and doesn't even attempt to address the question I asked.

If we observe a number, at minimum as a direction of 1d relative to time through time, the concept of groups does not necessarily contradict this as a number is merely a set of relations, a group another set of relations, etc. "Direction" is merely an observation of movement that exists as a localization where movement and localization are inseparable.

[Eodnhoj7]

Interesting.

High praise. Thank you. I make no claim of omniscience but I do try to be interesting.

So what happen in 1960? How Euclid formulate the problem makes no sense to me.

Edwin Hewitt figured out how to make infinitesimals logically rigorous in 1948. Abraham Robinson used Hewitt's idea to develop nonstandard analysis in 1961.

The point about Euclid is that he said a line is made up of points. But the philosophical question is, how can that be? Points are dimensionless and have size zero. Lines have both dimension and length. How do points, no matter how many, become a line?

A argue this elsewhere, along time ago:

1) A theoretical 1 dimensional point exists that is directed into itself ad-infinitum.
2) The 0d point is the limit of this 1d point as a dual no-dimensional structure.
3) (To skip a few steps). The 1d point exists in the 0d point and the 0d point inverts the 1d point into a 1d "extradimensional" (not intradimensional) structure by approximating the unity of the 1d point into multiple relative structures as lines.

[Justintruth]

....Infinitely many infinitesimals are summed to produce an integral."

I thought that the integral was the limit of a series of sums. I don’t think the integral is a sum outright but rather if you take a series of a particular type of sum and then take the limit of that series you get the integral

[wtf]

....Infinitely many infinitesimals are summed to produce an integral."

I thought that the integral was the limit of a series of sums. I don’t think the integral is a sum outright but rather if you take a series of a particular type of sum and then take the limit of that series you get the integral

Yes exactly. The concept of the limit replaces, or finesses if you will, the vague idea of the infinitesimal. There are no infinitesimals in the real numbers and there are no infinitesimals in so-called "infinitesimal calculus." Many many students come away from their calculus class confused on this point. An integral is NOT the sum of infinitesimals (even though engineers and physicists like to think of it that way!) As you note, an integral is the limit of Riemann sums as the partition size goes to zero.

The key idea with limits is that we replace the old idea of "infinitely small" or "infinitely close" with the idea of arbitrarily close or small. It's an idea that took over 150 years to work out. These subtleties are not taught to calculus students, since the assumption is that most students of calculus are not math majors but rather people who need to apply the methods of calculus, and not dive into the philosophical and mathematical underpinnings of the subject.

[Eodnhoj7]

....Infinitely many infinitesimals are summed to produce an integral."

I thought that the integral was the limit of a series of sums. I don’t think the integral is a sum outright but rather if you take a series of a particular type of sum and then take the limit of that series you get the integral

Yes exactly. The concept of the limit replaces, or finesses if you will, the vague idea of the infinitesimal. There are no infinitesimals in the real numbers and there are no infinitesimals in so-called "infinitesimal calculus." Many many students come away from their calculus class confused on this point. An integral is NOT the sum of infinitesimals (even though engineers and physicists like to think of it that way!) As you note, an integral is the limit of Riemann sums as the partition size goes to zero.

The key idea with limits is that we replace the old idea of "infinitely small" or "infinitely close" with the idea of arbitrarily close or small. It's an idea that took over 150 years to work out. These subtleties are not taught to calculus students, since the assumption is that most students of calculus are not math majors but rather people who need to apply the methods of calculus, and not dive into the philosophical and mathematical underpinnings of the subject.

Couldn't it be argued that infinity acts as a limit in itself, considering it provides the boundary between what is determinate and what is not?

This will build off of the point in regards to fractals, atomic facts (fractions of a fact which in themselves are facts).

If I am observing 1 → 2 and "→" in itself as a set of relations it "could" be observed as:

1.1→(((1.11→1.111→ ∞)→(1.12→1.121→ ∞)) → 1.2 → ((1.21→1.211→ ∞)→(1.22→1.221→ ∞)) → 1.3 → ∞) → 2

In this we can observe a series of numbers whose limits are determined by the indeterminate "infinity".

Now if this series is reverse we get a separate set of relations as evidenced below, considering the series is not defined except by it beginning and middle:

2 ← 1

1.9 ← (((1.91 ←1.911 ← ∞) ← (1.92 ←1.921 ← ∞)) ← 1.8 ← ((1.81 ←1.811 ← ∞) ← (1.82 ← 1.821 ← ∞)) ← 1.7 ← ∞) ← 1

"→", "←", and "()" exist as a triad in themselves. In these respects that while "1 → 2" and "2 ← 1" are fundamentally the same the relations which compose "→" and "←". They are dependent upon the point of origin as a boundary of limit considering both "→" and "←" are a set of relations in themselves which reflect through each other as further Infinity, considering in the context provided infinity is indeterminate and acts as a limit to the series in the respect it negates the continuity of definition. In simpler terms the series as movement/tending are dependent upon observing:

1) The point of origin and movement/tending leading to infinity as the end, or
2) The point of end and movement/tending leading to infinity as the beginning, or
3) The beginning and the end with the movement/tending as infinity.
4) All of the above at the same time in different respects.


"→" can be observed in the respect that 1 is pushing towards 2.
"←" can be observed in the respect that 1 is pulling toward 2.

No matter the manner you want to view it, the dual set of series maintain an inversive symmetry, however because the relations which compose these series are compose of infinite series neither set of relations are entirely the same. This applies directly to a logical argument itself considering "A → B" and "B ← A" are compose of atomic facts which require an infinite series of definition to maintain. This infinite series of definition does not just stem to the meta-logic which determines the nature of the series but the actual definitions of the words which are composed of further atomic definitions in the respect that one word is defined through another word ad-infinitum. So a "fractal" corresponds directly to a "part" or "atomic" and the dualistic qualitative and quantitative natures are tied together by "infinity". This infinity as fundamentally "limitless limit" synonymous to indeterminate gives a neutral ground from which quality and quantity expand as a series provides the boundary for both quantity and quality.

Now is this the logic you are going to learn in textbooks? My opinion is "ehhh...probably not", but it stems to inherent problems of supertasks, infinite series, intuitionist logic, etc. which are bound within the meta-logic itself....and keep in mind this is "my" argument so take it with a grain of salt.

This argument extends into a form of hyper-relativism and causes an extreme progress in definition at the expense of an obscurity induced complexed considering you can look at the same thing from two different angles and get completely different realities that stem paradoxically from the same reality. Inversion in direction leads to a multiplicity of definition and an effect of individuation.

[wtf]

This argument extends into a form of hyper-relativism and causes an extreme progress in definition at the expense of an obscurity induced complexed considering you can look at the same thing from two different angles and get completely different realities that stem paradoxically from the same reality. Inversion in direction leads to a multiplicity of definition and an effect of individuation.

Carl Friedrich Gauss couldn't have phrased it any better.