as I argued with TIDF earlier in the thread. There are many ways to determine a quantity without referencing an ascending order. — Metaphysician Undercover
Thanks for the clarification fishfry, but here's a couple more things still to clear up.
To me, the following statements contradict each other.
But the set of natural number may nonetheless be ordered in many alternative ways.
— fishfry
But no set has order. That's the axiom of extensionality. Will you kindly engage with this point?
— fishfry
Which is the case, no set has order, or a set may be ordered in many different ways. Do you apprehend the contradiction? Which is it, ordered in different ways, or not ordered? — Metaphysician Undercover
Let me go back to my question from the last post. What exactly constitutes "the set"? Is it the description, or is it the elements which are the members of the set. — Metaphysician Undercover
If it is the description, or definition, then order is excluded by the definition. — Metaphysician Undercover
But if the set is the actual participants, then as I explained already they cannot exist without having an order. If the supposed participants have no existence then they cannot constitute the set. — Metaphysician Undercover
That's why I ask, which is it? Can a set be ordered, or is it inherently without order? Surely it cannot be both. — Metaphysician Undercover
Let's look at "concept" as a noun, as if a concept is a thing. Do you agree that a concept is the product, or result of conception, which is a mental activity? There's different mental activity involved, understanding, judgement, conclusion, and effort to remember. Would you agree that the effort to remember is what maintains the concept as a static thing, So if a "concept" is used as a noun, and is said to be a thing, it is in the same sense that a memory is said to be a thing. Would you agree that if a mathematical concept is "a thing", it is a thing in the same sense that a memory is a thing? — Metaphysician Undercover
You assume that numbers are objects but argue that numbers are not objects? Sounds about right given your confusion. — Luke
By the axiom of extensionality, a set is entirely characterized by its elements, without regard to order. So the set {a,b,c} is the exact same set as {b,c,a} or {c,b,a}. — fishfry
Now we want to layer on the concept of order. To do that, we define a binary relation, which I'll call <, and we list or designate all the true pairs x < y in our set. So for example to designate the order relation a,b,c, we would take the base set {a,b,c}, and pair it with the set of ordered pairs {a < b, a < c, b < c}. Then the ordered set is designated as the PAIR ({a,b,c,}, {a < b, a < c, b < c}). I hope this is clear. — fishfry
The basic takeaway is that a set has no inherent order. We impose an order on a set by PAIRING the set with an order relation. — fishfry
A set is a collection of elements, regarded as an individual thing, a set. — fishfry
Perhaps it's the distinction between a bunch of athletes and a team, or a collection of birds and a flock. I'm sure some philosophers have found ways to describe this. A set is a collection of elements, along with the concept of their set-hood. That's the best I can do! — fishfry
A set is inherently without order and without any kind of structure. — fishfry
A set is entirely characterized by its elements; but a set is more than just its elements. It's the elements along with the collecting of the objects into a set. — fishfry
Do you understand the meaning of the word "if"? — Metaphysician Undercover
You also told us that you assume numbers are objects. — Luke
It's not that sets don't have orderings. It's that sets have many orderings (though in some cases we need a choice axiom or an axiom weaker than choice but still implying linear ordering). So the point is that there is no single ordering that is "the ordering". — TonesInDeepFreeze
Notice, the quoted passage says numbers are assumed when "you" count. And, it's your count that I argue is false. . — Metaphysician Undercover
Your "ascending order" is based on quantity, therefore your supposed "count" of ascending order means nothing unless it is determining a quantity. This is why "numbers" as objects are assumed, so that when you count up to ten you have counted ten objects, (numbers). — Metaphysician Undercover
And, if numbers are not true objects, as I argue is the case, then this is not a true act of counting at all. — Metaphysician Undercover
If we assume that a set necessarily has an ordering — Metaphysician Undercover
by what principle can we say that each of these many possible orderings constitutes the same set? — Metaphysician Undercover
What type of entity is an "element", such that the identity of a unity of numerous elements is based solely in the identity of its parts with complete disregard for the relations between those parts? — Metaphysician Undercover
Isn't this a sort of fallacy of composition? — Metaphysician Undercover
We prove from axioms. — TonesInDeepFreeze
If an axiom is false then the proof is unsound — Metaphysician Undercover
If an axiom is false then the proof is unsound. — Metaphysician Undercover
That "elements" may exist without an order is the falsity I've explained to you already. And if we say that "element" indicates an abstraction, then it is a universal, not a particular, and to assume that an abstraction is a particular is a category mistake. — Metaphysician Undercover
Let's assume a special type of "element", created, or imagined specifically for set theory. This type of element can exist in a multitude without that multitude having any order. Each of these elements would have no spatial or temporal relation to any other element, or else there would be an order, according to that relation. We could say that they are like points, but without a spatial reference, so that we cannot draw lines between them etc., because there is no order to them. But if they were like points, without spatial relations constituting order, there would be no way to distinguish one from another — Metaphysician Undercover
Unlike points though, there is something which distinguishes one element from another, so that in the set of (a,b,c,), "a" does not represent the same thing as "b" does. Can I conclude, that the distinct elements are separated from one another, and distinguished one from another, by something other than space? To make them distinct and individual, they must have separation, but the separation cannot be spatial or else they would have an order, by that spatial relation.
Do you see, that from this premise alone, we cannot give any order to any set? To give a set an order would be a violation of the fundamental meaning of "element" which allows that elements can exist as particulars without any spatial temporal; relations. To be able to talk about an order within a set, would require that we transform the elements into something other than "elements", something which could have spatial or temporal relations and therefore an order. Remember, even quantity requires spatial-temporal separation between one and the other, to distinguish separate individuals. — Metaphysician Undercover
No, sorry, it's not clear at all. You have imagined distinct "elements" which exist without any spatial or temporal relations, thereby having no order, though they are somehow distinct individuals. — Metaphysician Undercover
Now you want to add order. You have already defined order out of the set, to add it in, is blatant contradiction. — Metaphysician Undercover
What I need, is a clear explanation of what an "order relation" is. — Metaphysician Undercover
What type of relation are you attempting to give to these elements, which gives them an order, when you've already stipulated the premise that they have no order? — Metaphysician Undercover
The point is, that to give them existence without order requires a special conceptualization which I described above. Now if we want to proceed with that conceptualization, and now bring in principles of order, we must do so in a consistent way. So, we need to describe what separates one element from another, since it's clearly not space, and what makes it distinct as an element, in terms which do not give it a relationship to the others, to allow that the multitude of them do not already have any order, Then we need a principle by which order can be initiated within this non-ordered type of separation. — Metaphysician Undercover
Is it possible for an axiom to be false? Please explain. Don't refer to inconsistency. :roll: — jgill
Which axioms of finite set theory do you think are false? — TonesInDeepFreeze
I think you are just not cut out for mathematical abstraction and should pick another major. — fishfry
Enough. You win. You wore me out. — fishfry
Pick another major. — fishfry
You may well have a philosophical point to make, but you are preventing yourself from learning the subject. And it's learning the subject that would allow you to make more substantive rather than naive and obfuscatory objections. — fishfry
What else could demonstrate falsity other than a reference to some form of inconsistency?. — Metaphysician Undercover
An axiom is expressed as a bunch of symbols, so it must be interpreted. — Metaphysician Undercover
If in interpretation, there is a contradiction with another principle then one or both must be false. — Metaphysician Undercover
Notice there is an exchange of "equal" and "same" — Metaphysician Undercover
As I've argued in other threads, if we adhere to the law of identity, this is a false use of "same". — Metaphysician Undercover
I had difficulty even in grade school — Metaphysician Undercover
will you simply assert that mathematics is far superior to philosophy — Metaphysician Undercover
How do you propose that one proceed toward "learning the subject", when the most basic principles in that subject do not make any sense to the person? — Metaphysician Undercover
You seem to have a very naive outlook. How do you propose that one proceed toward "learning the subject", when the most basic principles in that subject do not make any sense to the person? — Metaphysician Undercover
I had difficulty even in grade school, when the teachers insisted on distinguishing numbers from numerals. Where are these "numbers" that the teacher kept trying to tell us about, I thought. All I could see is the numerals, and the quantity of objects referred to by the numeral. But the teacher insisted no, the numeral is not the number. So it took me very long to figure out that the numeral was not the "number" which the teacher was talking about, and that the number was just some fictitious thing existing in the teacher's mind, so I shouldn't even bother looking for it because I have to make up that fiction in my own mind, for there to be a number for me to "see". — Metaphysician Undercover
To me, the distinction between a numeral and a number is fundamentally unintelligible, as a falsity, because it requires producing a fictitious thing in my mind, and then talking about that fictitious thing as if it is a truth. — Metaphysician Undercover
Or will you simply assert that mathematics is far superior to philosophy, — Metaphysician Undercover
then run off and hide under some numbers somewhere when the unintelligibility of your principles is demonstrated to you? — Metaphysician Undercover
at first, the subject makes no sense — fishfry
It's like saying that learning to play a musical instrument is tremendously difficult at first so people should just give up. — fishfry
It's true of virtually EVERYTHING that at first, the subject makes no sense. You just do as you're told, do the exercises, do the homework, do the problem sets without comprehension, till one day you wake up and realize you've learned something. It must be that you've learned nothing at all in your life, having given up the moment something doesn't make immediate sense to you. — fishfry
When you learned to play chess, or any game -- bridge, poker, whist -- do you say, "Oh this is nonsense, no knight REALLY moves this way," and quit? Why can't you learn a formal game on its own terms? If for no other reason than to be able to criticize it from a base of knowledge rather than ignorance? If you've never seen a baseball game, it makes no sense. As you watch, especially if you are lucky enough to have a companion who is willing to teach you the fine points of the game, you develop appreciation. Is that not the human activity called LEARNING? Why are you morally opposed to it? — fishfry
Finally, even your basic objection to unordered sets is wrong. Imagine a bunch (infinitely many, even) of points randomly distributed on the plane or in 3-space. Can't you see that there is no inherent order? Then you come by and say, "Order them left to right, top to bottom." Or, "Order them by distance from the origin, and break ties by flipping a coin." Or, "Call this one 1, call this one 2, etc." — fishfry
So just adopt the formalist perspective. There are only numerals and the rules for manipulating them. It's a game. What on earth is your objection? Were you like this when you learned to play chess? "There is no knight!" "The Queen has her hands full with Harry and that witch Meghan!" etc. Surely you're not like this all the time, are you? — fishfry
What is the inherent order of the points in this set? Can you see that the points are inherently disordered or unordered, and that we may impose order on them arbitrarily in many different ways? Pick one and call it the first. Pick another and call it the second. Etc. What's wrong with that? — fishfry
I look at truth as corresponding with reality. — Metaphysician Undercover
If mathematics requires self-deception — Metaphysician Undercover
how to count, and simple arithmetic, addition, subtraction multiplication, division made sense to me right from the start. It was only later, when they started insisting that there existed a number, distinct from the numeral, that things started not making sense. — Metaphysician Undercover
If you think you can interpret the rules as we go — Metaphysician Undercover
The inherent order is the exact spatial positioning shown in the diagram. — Metaphysician Undercover
That about sums it up. Math is like religion, a whole bunch of bullshit which we are told to accept on faith — Metaphysician Undercover
I found that out at about tenth grade, despite living in an extremely mathematically inclined family. — Metaphysician Undercover
Math is like religion — Metaphysician Undercover
Again I don't agree with this. Many things I've learned made sense to me right from the start. Even learning the numerals, how to count, and simple arithmetic, addition, subtraction multiplication, division made sense to me right from the start. It was only later, when they started insisting that there existed a number, distinct from the numeral, that things started not making sense. — Metaphysician Undercover
I had a similar experience later with physics. We learned basic physics, then we learned about waves, and got to experiment in wave tanks. We learned that waves were an activity within a medium and we were shown through diagrams how the particles of the medium moved to formulate such an activity. All of this made very much sense to me. Then we were shown empirical proof that light existed as waves, and we were told that light waves had no medium. Of course this made no sense to me. — Metaphysician Undercover
When I learn a game, I must learn the rules before I play. If the rules are such that I have no desire to play the game, I do not play. It's not a question of whether the game makes sense or not, so the analogy is not a good one. — Metaphysician Undercover
Finally, you decided to address the issue. If there are points distributed on a plane, or 3d space, the positioning of those points relative to each other is describable, therefore there is an inherent order to them. If there was no order their positioning relative to each other could not be described.. — Metaphysician Undercover
You say that they are "randomly distributed", to create the illusion that there is no order. — Metaphysician Undercover
But the fact is that they must have been distributed by some activity, and their positioning posterior to that activity is a reflection of that activity, therefore their positioning is necessarily ordered, by that activity. — Metaphysician Undercover
If you think you can interpret the rules as we go, then I'd advise you not to play any games with me. — Metaphysician Undercover
The inherent order is the exact spatial positioning shown in the diagram. If any point changes location, then the order is broken. Is that so difficult to understand? A spatial ordering is not a matter of first and second, that is a temporal ordering. — Metaphysician Undercover
I learned to play a musical instrument, and it always made sense to me, right from the start. — Metaphysician Undercover
That the subject at first makes little sense is probably usually true. And it's true for me for many different subjects. But symbolic logic is one subject that made perfect sense to me immediately. — TonesInDeepFreeze
If mathematics requires self-deception, then this does not make sense to me, and so I will not proceed. — Metaphysician Undercover
Where are these "numbers" that the teacher kept trying to tell us about, I thought. All I could see is the numerals, and the quantity of objects referred to by the numeral. But the teacher insisted no, the numeral is not the number. So it took me very long to figure out that the numeral was not the "number" which the teacher was talking about, and that the number was just some fictitious thing existing in the teacher's mind, so I shouldn't even bother looking for it because I have to make up that fiction in my own mind, for there to be a number for me to "see". — Metaphysician Undercover
Even learning the numerals, how to count, and simple arithmetic, addition, subtraction multiplication, division made sense to me right from the start. It was only later, when they started insisting that there existed a number, distinct from the numeral, that things started not making sense. — Metaphysician Undercover
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