Excellent video. Reducing the views of Frege/Russell, JVN, and Cantor to fit into a 13 minute presentation is remarkable even if one does a bad job; to have done a fine job of it, as this presenter has done, is spectacular. Really well done. The ATM analogy was great. Distilled so much information.
While Cantor's approach, like the others, appears at first blush to be a beautiful solution, it does not in the final analysis capture the "two-ness" of pairs, but simply boils down Frege's set of all sets to arrive at what it is that makes them pairs. And that is their "two-ness". When we note that the set of all pairs, or the set {0,1}, or the set of 2 things about which nothing can be said except that together the two of them constitute the number 2 have a "two-ness" about them, which provides the appeal of these theories, none of them can explain how we already are able to recognize that "two-ness" without appeal to any set. The "two-ness" seems to be logically prior to all this set positing. Some people abhor the idea that numbers have existence independently of other objects or acts of counting, and I don't know why.
The last option, the Cantor option, is the worse one. It effectively defines a number in terms of itself. For example, if a person did not have a concept or definition of the number 2 to begin with, then to define the number 2 for that person, the requirement is to choose any 2 units. But the definition of 2 is unknown to that person, so he cannot choose 2 units, and therefore cannot apply the definition. In other words, the definition is circular. Also, the term "unit" is undefined. The best option is the one that mathematicians use, which is the second option. The disadvantage that the speaker mentioned for the second option is overcome by using one-to-one correspondence between the set {0, 1} (i.e. 2) and any set of two elements. A similar method can be used for each other number.
Don't the Frege-Russell and Cantor definitions both assume a concept of number in their definitions? The von Neumann definition doesn't seem to have that flaw, but it just gives a recursion series that doesn't seem to have anything specifically to do with number, as if it was a given constraint that any definition of number has to be given in terms of sets.
I may be misunderstanding your point, but it seems to me that what the Cantorian definition of number assumes is "the unit". The unit however isn't itself a number.
Re: the von Neuman idea, it seems clearly wrong to say a number IS the set of its predecessors, and I suspect it would not even seem to have any explanatory force if not for the fact that the set of the predecessors of 3, e.g. is a set with cardinality 3, which is all that Frege wanted for membership in his set of all triplets.
I doubt that Von Neuman himself would have ever said that a number IS the set of its predecessors. I think rather, he would have said that a number is ILLUSTRATED BY the set of its predecessors.
Sorry for my english, but I have to make a question: I am very perplexed about the process of abstraction depicted in this talk, as the remotion of all the differetiating features of two objects: if is true that, removing all the differetiating features of two object A and B, I obtain two 'bare' objects, a and b, called 'units', are these two units different one from another or are the same? Maybe they are different, otherwise {a,b} = {a,a} = {a}. But then, if I take another two object C and D, different from A and B, I obtain, by abstraction, two other units, c and d; what are the relation between (a,b) and (c,d)? c = a and d = b? Or c = b and d = a? Or they are all different one from another? But if {a,b} is different from {c,d}, then I find two different numbers two, namely {a,b} and {c,d}. This is strange. On the other hand, if any couple (not ordered) of objects A e B provides always the same couple (not ordered) of units, {x,y}, maybe we have to explain why this is so, and maybe it is not easy without risk of circularity.
But surely, the statement "two is the set of two units" is not a definition but a tautology. I don't see how it helps us to actually understand the concept of "two". It seems to me that, before you try to understand the concept of "two" or "three", you have to understand what we mean by "one". In other words, by virtue of what process do we determine there to be one of something? It must ultimately come down to how we understand difference/separateness, since we perceive "one" thing as being somehow differentiated/separate from the rest of reality.
which lies in how we operate our recognition, the way we recognize holds the notion of abstraction and therefore settheory in it, i guess. A tree is not just a bunch of atoms to our mind, but we have by evolution learned to recognize certain bunches of atoms as a seperate unit specified by the pattern that emerges from the arrangement of the atoms, the information inside their order, we could say color or spatial properties emerge by their interpretation.. could possibly be, that there is only unordered information around us, and we impose meaning to the abstract information structures that emerged by recognition, or we just read the real intrinsic structures off from things, the latter would correspond more with a "physics leads to properties" worldview and the former more to "perception and recognition state properties".
I doubt Russell developed the first formalization independently of Frege. Russell was heavily influenced by Frege, many things he just copied from him (like Frege's predicate logic) with a little modification.
Ive recently been studying the ancient Greek concept of 'arithmos', conventionally translated as 'number', though the two are apparently significantly different concepts. According to Paul Pritchard (Plato's Philosophy of Mathematics, 1995) the Greeks understood arithmoi as something like sets of abstract units. Numbers by contrast, at least as conceived by Descartes, Newton, and other early modern theorists, are something like abstract quantities or ratios. Modern numbers are taken to be unique individuals, ancient arithmoi were understood as multitudes or collections-or, if you like, as sets. It's striking to me that Cantor's view (at least as presented by Fine here) seems to be something of a return to the Greek concept of arithmoi. But my knowledge of Cantor is basically zero. Can anyone confirm whether this connection (between Cantor's definition of number and the ancient concept of arithmos) is correct?
There is a lot like this I would not be shocked. Free will was first mentioned with traction for me by Titus lubricious carious * (around the time of Plato) spelling may be iffy. But he mentioned how particles would have to have a force onto them that he called free will and said if particles have it then we must get it from a similar source. I will look into”arithmos” thank you because I have been trying to understand what is a number! So that’s an earlier reference for me. Thank you!
Modern Number system and roman number have different origins. In 724AD zero came to its existence in era of emper harshawardhan ( now India), by mathematician aryabhatt and this concept of idea zero and counting number was promoted by arab businessman from India to the rest of the world because that numbers were easy to use ..
Not being a mathematician myself, I've always been bothered that a number can, on the one hand, be used to represent a thing while, on the other hand, it can be used to show a relationship. For example, in 2+2 = 4, the "2" is a representation of a countable thing (2 apples + 2 apples). However, in an equation like 10/5 = 2, the "2" defines the relationship between the number 5 to the number 10 (5 goes into 10 two times). It seems (to me) logically inconsistent to say 3 - 1 = 10/5, because the result on either side of the equation is not the same "type" of 2. It's akin to that particle/wave duality in physics, where one measure represent a countable thing (a particle), while another measures the relationship (wave).
No. There are no "types" of 2. 2 is 2. All 2s are 2. In fact, the relationship between 5 and 10, expressed by your equation is embedded in the concept of the number 2. Think about it this way, "division" asks us how many sets of 5 are there in 10. The answer is 2 sets of 5. So, the answer '2' is still the same as is '2 apples', but instead of apples we have 'sets of 5'.
You've mixed numbers with relationships. The equation 3-1=2 MODELS the action of taking 1 apple for 3, but does but explain it, think of maths as a concept and the real world as a bi-product which by chance happens to follow its laws.
@@Human_Evolution- It is a deep question bro, I’ve still got mathematical (as well as philosophical) questions I’m still thinking pondering; in the grand scheme things, we know very little.
It surprises me that anyone should 'THINK' numbers are real. Here's my conclusion after years of questioning: As with words, 'numbers' are merely conceptual labels. And, like words, numbers represent a predetermined meaning, a 'value' that we attach to real or imagined objects. In the same way that a sentence is the mental processing of words, mathematics is also purely, the MENTAL PROCESSING of numbers. Therefore, it should be self-evident that universe cannot possibly have been 'created' by a MENTAL PROCESS... ...Or could it?
The argument contra the first view posed seems wrong, because it assumes if set A is a member of set B, then each individual member of A is also a member of B qua individual member of A. The number 2 was defined as the set of all pairs, but 2 is not a pair; it is a number. This seems like a sort of disguised third man argument, which does not seem compelling contra the theory of Socratic Forms, nor in this case either.
There´s no contradiction in set of number 2 "containing itset". When we know the number is a logical entity as it is a "set", and we know that the containing "process" is also a logical thing meaning subsumtion and reference, the problem is solved. So, Cantor´s sets are not ontologically different´s from Russell´s set. The difference between them is only the nature of the elements refereed. Fine is right about the nature of numbers. But numbers, as logical entities, lead us to accept that they can refer to any kind of signs (icons, symbols or indexes) without being confused with them.
Cantor's proposition is equal to Frege-Russell's. The so called "units" are just another way to refer to pairs. In Cantor's way we would take the set {1,2} and take away their characteristics, 1 is no longer 1 and 2 is no longer 2, in that way "making pairs". But it does not change anything because neither "units" nor "1" and "2" exist in nature, in that sense, Cantor is just changing nomenclature.
Try Piano`s Numbers in order to understand what numbers are. Alas, it is far from easy, since one has to derive the idea of numbers without reference to counting.
Regarding the unit theory of numbers, surely the two units cannot be distinguished, as they do not differ in any way? So, it might not be possible to distinguish the set of two units from the set of one unit or three units, and so on.
Im way over my head here, but i feel like the units have to be obviously distinguishable as thats what makes them 2 otherwise you wouldnt know whether it was two or just two ones if that makes any sense. My brain immediately went to dimenisions, like left and right in a 2D space. I wonder what the maximum number you can have of distinct units are before they are just repititions of multiple different numbers, im gonna guess 5 but thats only because im trapped thinking about dimensions with them being up down left right and time. I realise this isnt about dimensions but im desperately trying to tie them to the fabric of universe somehow as I feel they must be or we wouldnt be able to make such precise measurements.
Fractions cant be equaled but are counted as numbers solving this question by adding variables of infinate possibites to a problem formulated to divide and subtract
Units are defined as "bare distinguishables", there is nothing that can be said about a unit, other than that it is not some other unit. So there is no way to differentiate between two sets of two units as everything we can say about them is that they are not the same unit and that can be said about any two units. By a principle that says that if two things cannot be distinguished in any way at all, they must be one and the same thing, we conclude that there is only one set of two units.
What he describes are just competing views informed by different philosophical values. Frege/Russell takes the world to be undifferentiated facts, so is troubled by hierarchy. Von Neumann posits a hierarchy, so is axiological. Cantor tries to eliminate features of the world altogether via abstraction. Perfectionist and so appealing to many mathematicians but unrealistic. This is more about metaphysics than number. My view: numbers per se are abstract and so are never found by themselves in the world but are used by us to describe and manage facts and values in the world. Human values such as money (a possessive relation, not an enumeration of stuff although the discreteness of dollar bills and coins misleads us all) integrate facts such as are presented by human work (or materials used in work) via economic valuations. It is pointless to choose between the three views of number that he speaks of, as they are all present to us in the world simultaneously. Nicolai Hartmann had a better and more complete picture although the world finally resists all analysis and definitive answers.
This video is exactly the problem with teachers who have never understood what it means to be a number. The set theoretic approach confuses students and they never acquire a good understanding. The approach of the presenter was circular in the definition of 7. He didn't even bother trying to define the unit, which in his case is the dot he used. Also, he did not start with the concept of magnitude (which is not a number), but moved straight into combinations which presumes the student already understands numbers. *A number is the measure of a magnitude.* Let's see how to derive the rational numbers in just a few steps. 1. We begin with magnitude: A magnitude is the idea of size, dimension or extent. 2. We compare two magnitudes (let's use distances for all our magnittudes) and if we can tell they are not same, then all we can do is tell which is longer or shorter, but not how much longer or shorter. This is the first type of measure and is called qualitative measure. Example: A _ B C _____ D The second line is longer, but unless we have numbers, we can't tell the difference between the line lengths. Such a comparison is called a ratio of magnitudes and written as AB : CD (literally, Ab compared with CD) 3. Suppose we compare equal magnitudes. This leads to the idea of the unit, since equal magnitudes measure each other exactly, that is, the difference is 0 and the result is that only one (unit) such distance is required to measure the other. So, we can choose the unit to be any magnitude we desire and write it as U : U. 4. Now, if we have two magnitudes that are measured exactly by the unit, then we can tell the difference in units, that is, we have quantitative measurement. At this stage we have 0, 1, 2, 3, etc, that is, natural numbers which are ratios of unit multiples to the unit. 5. Next, we have the idea of fraction to describe parts of the unit. This is a ratio of natural numbers to other natural numbers. These numbers are called rational numbers. 6. Finally, there are those magnitudes that cannot be measured. These have no common measure with the unit or parts thereof and are called incommensurable magnitudes. Examples are pi, sqrt(2), e, etc. These are incorrectly called irrational numbers, but they are not numbers at all because they cannot be measured by the unit. This is the perfect derivation of number which is independent of the human mind or any other mind. This is what Euclid was attempting to do in his Elements.
As you demonstrated in the phrase "we compare two magnitudes", the geometric conception of numbers presupposes the arithemtical (or set theoretical) conception of numbers. The idea behind your derivation is fully embedded in the set theoretical definition of rational numbers.
Presumably numberS are one more than number.. I don't know, is there any such thing as fiveness. Can there be numbers without memory? The real test is: Is it possible to remain awake in front of an interesting and alive question such as what are numbers without killing it with answers , or what is sometimes called 'Once-and-for-all thinking" or numbers are bla bla bla and that is the end of that. Maybe come at it from another angle and ask how exactly do I experience numbers or whatever is in question. I was forced on pain of a beating and other sanctions to learn what is called mathematics but never one in my near three quarters of a century have I had any need for mathematics beyond simple arithmetic, never used or needed an equation simultaneous quadratic or any other flavour in my entire life, but it is an alive question what are numbers is it not? I think I'll just keep it as a question, and it can go in the box with other questions such as what is time and what is understanding, what am I etc. why kill an alive question?
What makes you suppose that you *can* 'think'-assuming you have an idea what you mean by thinking? More to the point what exactly are you calling 'thinking'?
Sadly most of this talk is pure rubbish. Much of what he is presenting is "Naive Set Theory"... it is called "naive" for a reason. In truth, nobody even knows to this day if "sets" are nothing but fictions...even if they are so called "useful" fictions. Unless of course "sets" are just merely the symbols on a page...but that would then make numbers objects given to our senses which the speaker rejects. Indeed, Zermelo himself (a key founder of modern formal set-theory) saw the need to introduce the so called "empty set" as "fictitious". Why is that? "Naive set-theory" is still taught at universities pretty much everywhere even though it really is falsifying the true state of affairs. It also is usually asserted that set-theory is the foundation for ALL of mathematics...pure rubbish.There are now literally thousands of mathematical theorems probably in about every branch of mathematics which have been now shown to be independent of standard formal set-theory, i.e., "ZFC". Yet, there is little agreement on how a new "axiom" can even be selected since mathematicians can't even come to a basic agreement of what a modern "axiom" is supposed to be. When formal set-theory was introduced to get around the contradictions it was held that the new so called "axiomatic method" was ONLY justified if non-contradiction could be proven... this is now known to be impossible. His talk is titled "What are numbers" but only makes mention of kindergarten counting numbers... then claims that thankfully do to Cantor we now know what numbers are! What is the point of this talk? Does the speaker consider the "axioms" of set-theory which are needed to constructs the "real numbers" to be true? I doubt it...yet now after his little talk we are supposed to thankfully know what numbers truly are!??? Mathematicians should be ashamed of themselves for not even making an attempt to be honest about the true state of affairs regarding their "foundational" set-theory...most don't even seem to care. The very discipline which traditionally was held as a standard of rigorous thinking, i.e., "the queen of the sciences", has fallen into a morass.
Professor Fine authored an Article titled “Vagueness, Truth and Logic,” which is widely thought to have profoundly altered the course of a debate that has been going on among philosophers for thousands of years. He is responsible for the altering of a view held by Philosophers since the time of Aristotle with the above mentioned work. Why is that not exciting to you?
Ryan Ortega Oh did he, really? Author an article? Wow! And it's titled exactly like thousands of other analytic philosophy papers that appear on a daily basis and also claim to "profoundly alter a long-standing debate"? Why is it then that nobody outside a circle of professional philosophers living on taxpayers' money gives a damn? Anyway, I was talking about this TEDtalk, which is undeniably a dud. I imagine he must also have an article where he "rebuts" Frege's arguments in the introduction to Grundlagen in order to return to the obsolete Cantorian conception. Now that must be a real bore!
volodask Well, if you care a great deal about the value of Professor Fine's work, then you understand of course that it is highly technical and a TED Talk may not be the best place to expound too deeply on his work, but at least the information is getting heard by some who may not have otherwise known of the work.
