Isn't the definition of infinitesimal contradictory?

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Re: Isn't the definition of infinitesimal contradictory?

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Atla wrote: Sat May 26, 2018 7:23 am If we assume the Continuum hypothesis to be correct and so there is only one unique field of hyperreals, then would this uniqueness be somewhat similar to for example the uniqueness of the Monster group from group theory: similar in the sense that it would be a certain specific mathematical structure, found in higher mathematics, that's simply "there", for some reason?
Yes.
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Re: Isn't the definition of infinitesimal contradictory?

Post by Impenitent »

A_Seagull wrote: Sat May 26, 2018 5:10 am Infinitesimals don't exist.

Not even numbers exist.

All you have in mathematics are symbols and concepts that are associated with those symbols.

So the symbol '=' is no more than a symbol within the system of mathematics. And that symbol is typically associated with the concept of "equal" and all that entails.
extending symbolic agreement, but existence can be tricky

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Re: Isn't the definition of infinitesimal contradictory?

Post by Eodnhoj7 »

A_Seagull wrote: Sat May 26, 2018 5:10 am Infinitesimals don't exist.

Not even numbers exist.


All you have in mathematics are symbols and concepts that are associated with those symbols.

So the symbol '=' is no more than a symbol within the system of mathematics. And that symbol is typically associated with the concept of "equal" and all that entails.

In agreement with the purple, but from a seperate perspective.

As said before I cannot see an infinitesimal being anything other than a progressive series to point zero that in itself can only be "localized" in the respect of observing it as a series and not a finite number.

The paradox is that if I observe the infinitesimal as a specific series it becomes finite in the respect it is "x series of numbers". This is a localized boundary of movement, quantitatively speaking (although I doubt this terminology is what they use in the math community).
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Re: Isn't the definition of infinitesimal contradictory?

Post by wtf »

Atla wrote: Sat May 26, 2018 7:23 am
wtf wrote: Wed May 23, 2018 7:00 pm By assuming a weak form of the axiom of choice, one can show the existence of a gadget called a nonprincipal ultrafilter, which one can then use to produce a field (a mathematical system in which we can add, subtract, multiply, and divide) in which there are infinitesimals. This field is called the hyperreals. [Technically there are many such fields since different nonprincipal ultrafilters give rise to nonisomorphic fields of hyperreals. And it's a mathematical curiosity that the Continuum Hypothesis implies that there is only one unique field of hyperreals. As you can see we are in deep foundational waters].
Pardon me for my ignorant question (ignore it if it's just word salad): if we assume the Continuum hypothesis to be correct and so there is only one unique field of hyperreals, then would this uniqueness be somewhat similar to for example the uniqueness of the Monster group from group theory: similar in the sense that it would be a certain specific mathematical structure, found in higher mathematics, that's simply "there", for some reason?

(I just find it fascinating when mathematics seems to discover structures that are simply "there". I'm coming from the physical multiverse hypothesis angle; I have the impression that such structures may be somehow linked to the unique topology of our universe, but in ways we can't really understand yet. I also read somewhere that "set multiverse" ideas are a thing now in mathematics, and people who hold such views are more likely to think the the Continuum hypothesis may be correct.)
First, let's look at far simpler cases. Many abstract mathematical structures exist in the real world. For example take the set of permutations on three letters a, b, and c. We have: abc, acb, bac, bca, cab, and cba, six altogether. We can combine two permutations to get another (for ex. "swap the first two then swap the second two"), permutations are reversible, there's an identity permutation (abc) and composition of permutations is associative. So the six-element set of permutations on 3 letters is a group, an abstract mathematical structure. Group theory finds physical application in quantum physics and crystallography. So groups have some sort of actual existence even though you can't pick one up and put it under a microscope or smash it with a hammer.

Likewise, the hyperreals, whether you regard all the distinct hyperreal fields in the absence of CH, or the unique one in the presence of CH, are "out there" in some sense. What sense that is, I can't say. Perhaps a philosopher can explain to me what kind of existence the group of permutations on 3 letters has, then I'll know.

However in the case of the hyperreals, this existence is more murky than it is for elementary examples like groups. To construct the hyperreals, we need a weak form of the Axiom of Choice. So any such "construction" is actually nonconstructive. Nobody can visualize or write down any particular nonprincipal ultrafilter or its corresponding field of hyperreals.

So I have two philosophical questions. One is, what kind of "existence" does an abstract group have? And, what kind of existence does a nonconstructive mathematical object have? Note that there is a philosophy of mathematics, namely constructivism, that would accept the existence of the permutation group on 3 letters, but reject the existence of any nonconstructive object. So philosophers do discuss these issues.
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Re: Isn't the definition of infinitesimal contradictory?

Post by Eodnhoj7 »

wtf wrote: Sat May 26, 2018 5:41 pm
Atla wrote: Sat May 26, 2018 7:23 am
wtf wrote: Wed May 23, 2018 7:00 pm By assuming a weak form of the axiom of choice, one can show the existence of a gadget called a nonprincipal ultrafilter, which one can then use to produce a field (a mathematical system in which we can add, subtract, multiply, and divide) in which there are infinitesimals. This field is called the hyperreals. [Technically there are many such fields since different nonprincipal ultrafilters give rise to nonisomorphic fields of hyperreals. And it's a mathematical curiosity that the Continuum Hypothesis implies that there is only one unique field of hyperreals. As you can see we are in deep foundational waters].
Pardon me for my ignorant question (ignore it if it's just word salad): if we assume the Continuum hypothesis to be correct and so there is only one unique field of hyperreals, then would this uniqueness be somewhat similar to for example the uniqueness of the Monster group from group theory: similar in the sense that it would be a certain specific mathematical structure, found in higher mathematics, that's simply "there", for some reason?

(I just find it fascinating when mathematics seems to discover structures that are simply "there". I'm coming from the physical multiverse hypothesis angle; I have the impression that such structures may be somehow linked to the unique topology of our universe, but in ways we can't really understand yet. I also read somewhere that "set multiverse" ideas are a thing now in mathematics, and people who hold such views are more likely to think the the Continuum hypothesis may be correct.)
First, let's look at far simpler cases. Many abstract mathematical structures exist in the real world. For example take the set of permutations on three letters a, b, and c. We have: abc, acb, bac, bca, cab, and cba, six altogether. We can combine two permutations to get another (for ex. "swap the first two then swap the second two"), permutations are reversible, there's an identity permutation (abc) and composition of permutations is associative. So the six-element set of permutations on 3 letters is a group, an abstract mathematical structure. Group theory finds physical application in quantum physics and crystallography. So groups have some sort of actual existence even though you can't pick one up and put it under a microscope or smash it with a hammer.

Likewise, the hyperreals, whether you regard all the distinct hyperreal fields in the absence of CH, or the unique one in the presence of CH, are "out there" in some sense. What sense that is, I can't say. Perhaps a philosopher can explain to me what kind of existence the group of permutations on 3 letters has, then I'll know.

However in the case of the hyperreals, this existence is more murky than it is for elementary examples like groups. To construct the hyperreals, we need a weak form of the Axiom of Choice. So any such "construction" is actually nonconstructive. Nobody can visualize or write down any particular nonprincipal ultrafilter or its corresponding field of hyperreals.

So I have two philosophical questions. One is, what kind of "existence" does an abstract group have? And, what kind of existence does a nonconstructive mathematical object have? Note that there is a philosophy of mathematics, namely constructivism, that would accept the existence of the permutation group on 3 letters, but reject the existence of any nonconstructive object. So philosophers do discuss these issues.

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.
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Re: Isn't the definition of infinitesimal contradictory?

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-1- wrote: Fri May 25, 2018 9:01 pm
bahman wrote: Fri May 25, 2018 8:30 pm
-1- wrote: Fri May 25, 2018 8:20 pm
May I know why you're asking that?
You haven't add much yet. What I am looking for is a proper definition of infinitesimal if that really exist.
1. Under what authority do you feel you have the power to decide whether a definition is proper or not?

2. "if (the infinitesimal) really exists." Do you want a definition to something you have trouble even knowing if it exists or not?

Is it the definition of "infinitesimal" you need, or a tangible description of a vague concept that you have for "infinitesimal"?
Authority? I am just curious and looking for an agreement in understanding of subject matter.
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Re: Isn't the definition of infinitesimal contradictory?

Post by wtf »

Eodnhoj7 wrote: Sat May 26, 2018 7:10 pm 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.

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.
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Re: Isn't the definition of infinitesimal contradictory?

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A_Seagull wrote: Sat May 26, 2018 5:10 am Infinitesimals don't exist.
How about motion? How could have motion without infinitesimal?
A_Seagull wrote: Sat May 26, 2018 5:10 am Not even numbers exist.
Not objectively but we experience them as a result of interacting with reality.
A_Seagull wrote: Sat May 26, 2018 5:10 am All you have in mathematics are symbols and concepts that are associated with those symbols.
Yes, I agree that this is a good description of mathematics.
A_Seagull wrote: Sat May 26, 2018 5:10 am So the symbol '=' is no more than a symbol within the system of mathematics. And that symbol is typically associated with the concept of "equal" and all that entails.
But there sometimes reflect something in reality.
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Re: Isn't the definition of infinitesimal contradictory?

Post by wtf »

bahman wrote: Sat May 26, 2018 7:41 pm
A_Seagull wrote: Sat May 26, 2018 5:10 am All you have in mathematics are symbols and concepts that are associated with those symbols.
Yes, I agree that this is a good description of mathematics.
Isn't that rather nihilistic? When we say that 1 + 1 = 2, Mr. Seagull claims this is nothing but symbolic manipulation. But anyone else would note that this is a notation for something that we all perceive to be a truth about the world.

The doctrine of formalism is helpful in doing mathematical logic. But it's seriously deficient as an explanation for why mathematics is effective in the world.

The example I gave of group theory is on point. Groups are highly abstract and beginners generally have a hard time understanding why anyone cares about them. Yet we could not do modern physics without group theory; and geometry itself is now regarded as the study of those properties of space that are invariant under the actions of various groups.

Formalism fails as an explanation of anything.
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Re: Isn't the definition of infinitesimal contradictory?

Post by bahman »

wtf wrote: Sat May 26, 2018 7:53 pm
bahman wrote: Sat May 26, 2018 7:41 pm
A_Seagull wrote: Sat May 26, 2018 5:10 am All you have in mathematics are symbols and concepts that are associated with those symbols.
Yes, I agree that this is a good description of mathematics.
Isn't that rather nihilistic? When we say that 1 + 1 = 2, Mr. Seagull claims this is nothing but symbolic manipulation. But anyone else would note that this is a notation for something that we all perceive to be a truth about the world.

The doctrine of formalism is helpful in doing mathematical logic. But it's seriously deficient as an explanation for why mathematics is effective in the world.

The example I gave of group theory is on point. Groups are highly abstract and beginners generally have a hard time understanding why anyone cares about them. Yet we could not do modern physics without group theory; and geometry itself is now regarded as the study of those properties of space that are invariant under the actions of various groups.

Formalism fails as an explanation of anything.
I already mentioned that mathematics sometimes reflect something we observe in reality.
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Re: Isn't the definition of infinitesimal contradictory?

Post by wtf »

bahman wrote: Sat May 26, 2018 8:11 pm I already mentioned that mathematics sometimes reflect something we observe in reality.
Right. But Mr. Seagull appears to deny even that, if I understand his views correctly.
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Re: Isn't the definition of infinitesimal contradictory?

Post by bahman »

wtf wrote: Sat May 26, 2018 8:17 pm
bahman wrote: Sat May 26, 2018 8:11 pm I already mentioned that mathematics sometimes reflect something we observe in reality.
Right. But Mr. Seagull appears to deny even that, if I understand his views correctly.
So you believe that mathematics always reflect something in reality. I am not a mathematician but I guess one can find a mathematical entity which does not exist in reality.
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Re: Isn't the definition of infinitesimal contradictory?

Post by wtf »

bahman wrote: Sat May 26, 2018 8:24 pm So you believe that mathematics always reflect something in reality.
I didn't say that nor do I believe it.
bahman wrote: Sat May 26, 2018 8:24 pm I am not a mathematician but I guess one can find a mathematical entity which does not exist in reality.
Of course. We have non-Euclidean geometry and we have Euclidean geometry. Both are logically self-consistent but they can't both be true about the world. So at least one of them is a mathematical abstraction that does not exist in reality.

But mathematical formalism (which is what I'm getting from Mr. Seagull's position; and if I'm misunderstanding him I hope he'll correct me on this point) 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.
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Re: Isn't the definition of infinitesimal contradictory?

Post by Atla »

My current guess as a non-mathematician is that mathematics may be a system of human-made abstractions, but these abstractions are inherently "generated from" and "consistent with" the particular structure/workings of our universe. For example I find it fascinating that there are exactly 26 sporadic simple groups in group theory, and the largest sporadic group has exactly 196883 dimensions, and no one quite knows why. These seem like such arbitrary numbers, and yet they seem to follow directly from simpler mathematics. Could such numbers reflect something about the specific structure of our universe? Could a unique field of hyperreals reflect something about our universe?

(unanswerable questions at this point; sry for being a little repetitive I'm just rambling)
Last edited by Atla on Sat May 26, 2018 10:16 pm, edited 2 times in total.
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Re: Isn't the definition of infinitesimal contradictory?

Post by bahman »

wtf wrote: Sat May 26, 2018 8:28 pm
bahman wrote: Sat May 26, 2018 8:24 pm So you believe that mathematics always reflect something in reality.
I didn't say that nor do I believe it.
bahman wrote: Sat May 26, 2018 8:24 pm I am not a mathematician but I guess one can find a mathematical entity which does not exist in reality.
Of course. We have non-Euclidean geometry and we have Euclidean geometry. Both are logically self-consistent but they can't both be true about the world. So at least one of them is a mathematical abstraction that does not exist in reality.

But mathematical formalism (which is what I'm getting from Mr. Seagull's position; and if I'm misunderstanding him I hope he'll correct me on this point) 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.
So we are in the same page.

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.
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