Traffic laws are a made-up game too. — fishfry
But the word "contradiction" in mathematics has the meaning that I said: "A and not A" is not provable for any A. — Mephist
What you call "contradiction", the impossibility to identify the terms of the language with physical objects, is not considered as a problem in mathematics: it's simply ignored. — Mephist
The rules of deduction (used in proofs) should not depend in any way on the meaning (or correspondence to real physical objects) of the words. — Mephist
So, you say that this is all wrong, because you are allowed to create axioms that don't have any correspondence to reality. — Mephist
"Contradiction" has nothing to do with identifying things with physical objects, it relates to how words are defined. So for example, if "set" is defined as something having extension, and "empty set" is defined as a set having no extension, then there is contradiction here. "Empty set" breaks the rules expressed in the definition of "set", and therefore cannot be a set. — Metaphysician Undercover
OK, let's follow you definition of "set" (that is not the definition used in ZFC set theory, but we are considering an alternative definition because we do not accept MY mathematical rules). — Mephist
But now we are not finished yet: we have still to define what is "extension".
I think you have two possibilities:
1. define "extension" in terms of another property (something like "occupation of space"? I don't know..)
2. consider "extension" as an undefined "primitive" notion
- In case of 1. you end up in an infinite chain of definitions (of course you cannot define "extension" in terms of "a set", right?)
- In case of 2. you just did what today's mathematics do, just replacing the primitive notion of "set" with the primitive notion of "extension" and changing the definitions accordingly. — Mephist
But now what prevents me to consider a "null extension"?
"extension" at this point is an undefined notion, so "null extension" does not generate any contradiction now.
And if you allow "null extension", why not allow "empty set" and consider "set" to be a primitive notion instead? — Mephist
Following this argument that "null extension" is not allowed, you could say for example that a segment with "null length" is not allowed, so a point is not a segment. That is OK, but it's not due to a contradiction: it's only a choice of your definitions. Defining a point as a segment with no length does not create any contradiction, if you consider a segment as a primitive notion and a point as a derived notion. — Mephist
P.S. Try to take a look at Euclid's elements (https://en.wikipedia.org/wiki/Euclid%27s_Elements)
Here are the definitions, from Book 1 (taken from the book 'The elements of Euclid" by Oliver Byrne)
1. A point is that which has no parts
2. A line is length without breadth
5. A surface is that which has length and breadth only
Are these definitions contradictory? — Mephist
P.S. Try to take a look at Euclid's elements (https://en.wikipedia.org/wiki/Euclid%27s_Elements)
Here are the definitions, from Book 1 (taken from the book 'The elements of Euclid" by Oliver Byrne)
1. A point is that which has no parts
2. A line is length without breadth
5. A surface is that which has length and breadth only
Are these definitions contradictory? — Mephist
A set is a bottom-up conception, assembling a whole from discrete parts. True continuity is a top-down conception, such that the whole is more fundamental than the parts. — aletheist
The fact that you have to use intuitionistic logic with it's weird rules about double negation, to make sense of "d^2 = 0" even if "d =/= 0" in my opinion is just a "wrong" definition of the negation operator: it should be called "complement of" instead of "not" (for example we should say "d belongs to the complement of 0" instead of "d =/= 0"). — Mephist
"true continuity" can be defined even using standard set theory. Actually, even category theory can (and usually is) be based on standard set theory. — Mephist
the construction doesn't satisfy the field axioms — fdrake
Moreover, every function from the constructed number line with infinitesimals to itself turns out to be smooth (so, continuous and differentiable). If I've read right anyway. — fdrake
Why not? Which of the field axioms are not satisfied? — Mephist
Why is this a problem? — Mephist
You cannot divide by a number that has it's "base" part 0. That works at the same way as the usual real numbers — Mephist
All you need to do to turn any non-zero dividing element into a zero dividing element is to multiply it by d. So all elements zero divide. — fdrake
I appreciate your efforts to make sense of this for me. I am not just trolling. — Metaphysician Undercover
To define "extension" as a property which something can both have and have not, at the same time, is just a trick of sophistry, designed to dodge application of the law of non-contradiction — Metaphysician Undercover
I don't see this. I cannot see how you made the contradiction go away. All I see is a trick of sophistry, which hides the contradiction behind the illusion that zero extension is some sort of extension. — Metaphysician Undercover
A segment is defined by giving two points. if the two points are coincident (the same point), then it's the same thing as only one point. If the two points are distinct, the length is the measure of their distance from each-other. I don't see any problem with this definition.So let's look at this example of the segment and the point. You define "point" using "segment". A point is a segment without any length. So the property here is "length". The definition of "length" as you described with "extension" is irrelevant. But from my perspective we need to ensure that "length" is not a sort of property which a thing can both have and not have, at the same time, or else the definition of length would become relevant, as a sophistic trick. The subject, or category is "segment", a point is defined as a type of segment. So we now need a definition of "segment" to make sense of what a point is. Remember that we have just allowed for a segment with no length, so "length" cannot be a defining feature of "segment". How would we proceed to define "segment" now? — Metaphysician Undercover
I submit that this is a similar situation to what we have with "set". If we define "empty set", such that it is a real set which has no extension, then "extension" cannot be a defining feature of "set" without allowing that "extension" is a sort of property which defies the law of non-contradiction.. — Metaphysician Undercover
No I don't see any contradiction here. There is nothing to imply that a point is a line without length. That would be contradictory when #2 says that a line is length — Metaphysician Undercover
Let's look at it this way. The "line" introduces a new property which the preceding "point" has not, "length". The "surface" introduces a property which the preceding "line" has not, "breadth". But the "surface" also maintains the property of the "line", which is "length". Following this pattern, the "line" ought to maintain the property of the "point". But "no parts" is a sort of negation of a property, instead of a proper property. We can say that this negation is the property which the "point" has. So if we understand "no parts" as a negation of all properties, we'd have to understand the property of the point as "no properties", and this would be contradictory. It would be contradictory, to say that the property of a thing is that it has no properties. However, we do not understand "no parts" as "no properties", so the point is defined by what it does not have, and what it does have is left empty or undefined.
In the case of the "empty set", the property which it does not have is extension. However, being designated as a "set", it also has whatever property is proper to a "set". If extension is a defining property of a set, we have contradiction because we talking about a thing which is said to have extension (by the type of thing that it is said to be), yet it is also said to have no extension by the value given to that property. — Metaphysician Undercover
No, the whole point of talking about "true continuity" is to distinguish it from (analytical) "continuity" as defined in accordance with standard set theory. The real numbers do not possess true continuity, because numbers of any kind are intrinsically discrete. However, they serve as a useful model of continuity, adequate for most mathematical and practical purposes."true continuity" can be defined even using standard set theory. — Mephist
I am not a mathematician, but my understanding is that this is exactly backwards. Set theory can be established within category theory, but category theory cannot be established within set theory. "Set" is one of the categories, but there are others that need not and do not conform to standard set theory.Actually, even category theory can (and usually is) be based on standard set theory. — Mephist
I am not a mathematician, but my understanding is that this is exactly backwards. Set theory can be established within category theory, but category theory cannot be established within set theory. "Set" is one of the categories, but there are others that need not and do not conform to standard set theory. — aletheist
"null" is an attribute of "extension": an extension can be "null" or "not null". A set can be defined as something having extension (following your definition). null extension => empty set; not null extension => not empty set. Maybe this is a trick of sophistry, but it avoids the contradiction. — Mephist
A segment is defined by giving two points. if the two points are coincident (the same point), then it's the same thing as only one point. If the two points are distinct, the length is the measure of their distance from each-other. I don't see any problem with this definition. — Mephist
Yes, that's the same thing: you don't define what a set is, but just give some properties that any set should have, and one of the attributes of a set is the fact to be empty or not: this is just an attribute of any set: no need to define the word "extention". — Mephist
In my opinion to say "A line is length without breadth" is like saying "A line is a rectangle with zero width". He means that real objects have 3 dimensions, but a line is like a real object that has only 1 dimension. The other two dimensions are missing. — Mephist
Well, we could do the same with sets: adding properties instead of subtracting
- an "point-set" is a set with no parts
- a "line-set" is an extension of the "point-set" that introduces a new property: the number it's parts.
For me, that's the same logical construction. Why this should not be allowed?
At the end, they are all similar ways to do the same thing: attach some properties to an object to describe it without giving an explicit definition in terms of other objects! — Mephist
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