Post now completed.
Saw this on today's Wikipedia front page:
Quine–Putnam indispensability argument
https://en.wikipedia.org/wiki/Quine%E2%80%93Putnam_indispensability_argument
Expressed as this syllogism:
- We ought to have ontological commitment to all and only the entities that are indispensable to our best scientific theories.
- Mathematical entities (ME) are indispensable to our best scientific theories.
- Therefore, we ought to have ontological commitment to mathematical entities.
So in summary:
- The Quine–Putnam indispensability argument is not "indispensable to our best scientific theories."
- So we have no ontological commitment to it.
- If ontological commitment is necessary for validity then this invalidates the theorem.
- So the theorem represents an example of a valid argument to which we need no ontological commitment.
- This means that "ontological commitment" and existence has no bearing on the validity of an argument.
- So the Quine–Putnam indispensability argument still stands in that things indispensable to our best scientific theories may exist. But it also says (by contradiction) that valid things may also not exist. This makes existence a rather arbitrary and unimportant consideration.
In more depth:
Well Quine is a smart chap and a preeminent logician so should I really be arguing?
Anyway I smell a rat. He's an American and as a rule Americans seem unable to conceive of the Abstract preferring to force it into simple boxes like "existence" or "matter" or something rigid. If there is one limitation of US thinking: it is rigid.
Now Richard Feynman is on record as saying:
"I believe it is the atomic hypothesis (or the atomic fact, or whatever you wish to call it) that all things are made of atoms"
NOTE ON SRH:
However by SRH we know there is a contradiction by self-reference here. Let me note that the proof of SRH in the general case has not been shown yet. But it has been noted that the proof of the "existence of a contradiction by self-reference" namely SRH would require self-reference and so imply a contradiction of itself. We then showed that it may be possible if the SRH was a proof of the type "all rules have exceptions" this allowing for itself to be an exception.:
But if the statement "All self-reference implies the existence of a contradictory statement" is true then it itself implies there is a statement that contradicts itself. Which is a contradiction and so makes it true! Need think more carefully whether this allows for an escape clause.
Perhaps we could also argue that self-reference cannot occur in a contradictory statement for what is it referring to? Casual argument but things like that.
BACK TO FEYNMAN
So by SRH anyway obviously Feynman's statement is not true because if it is a thing then it is not made of atoms. So Feynman is either saying that:
- not everything is made of atoms (if he is presenting a contradiction) , or he is saying that
- the Atomic Hypothesis is not a thing.
So we are making a distinction between the:
- existence of things that we can sense and physically interact with, and the
- existence of the ideas about things in the mind.
Do mental objects exist? Well thoughts must exist because we can distinctly say when we have one and when we don't like "Eureka!" was Archimedes noting the occurrence of an idea. That which occurs must exist?
But then we have the instance of an idea, and the general idea that we may not be thinking about right now but we can think about any time we like after it is learned. That entry in Wikipedia that logs the nature of idea like a blueprint or program for anyone who wishes to think it. That Wiki page you would say exists in the sense it takes up disk space in California, and that can be copied into any number of other existence of data on machines around the world and displayed so people can read it. Indeed the question of digital copies does raise the question of what form the "copy write" on the original applies to. Yet Archimedes idea now exists whatever form it takes.
And the idea that "Abstract things exist" itself exists, at least according to Quine. Yet it is weird that while Scientific phenomena can be tested, why can't Quine's idea? If it exists like a scientific observation where is the observation? Quine must argue that because Abstract ideas are involved in science then they too must exist. They are clearly different from scientific theories themselves.
But does Quine's idea itself get involved in Scientific Theories?
And here is the proof he is wrong. No!
So Quines Abstract Idea that "Abstract ideas exist" does not itself have to exist. SO it stands as an example of an Abstract Idea that does not need to exist. Regardless whether Scientific Theories require the existence of existing theories, Quine idea does not and shows that this need not be true. So it totally defeats itself.
Indeed this is generally true. If we do not already know whether "abstract ideas" exist we cannot provide any abstract proof that they do. This is a "castle in the sky." If your foundation does not work, no mater how much you build on it is will not work.
Quine is trying to create an "Abstract idea" in support of "abstract ideas."
2 take homes:
- This is yet another demonstration of SRH. How many do we need before we see the general proof itself?
- Americans are generally stuck in a "narrowing" of thought. The American tendency to take short cuts, hammer things down, fix things in stone, make things simple and solid and saleable means they ultimately reject the very foundations they started with. The US needs to understand that if it is not already sufficient then no amount of work built on that foundation can be better.
Q.E.D.
.END
|
|
|
|
|
|
|
\ | /
V
Below are fragments when preparing this post. They include a partial investigation of the difference between things and idea; of general ideas, actual thoughts, and reality. Obviously thoughts have an actual existence as they take place in time and can be detected in the brain, but they are "about" abstract things. So we could have the idea about an idea. As I am doing here. They are not the same. Like a "name" is not the same as the "thing". In here lies fuel for SRH.
So the problem I'm looking for is a 3rd Man Argument type regress.
Let us use the abstract mathematical concept of a Circle.
C = ∀x : |x-P| = R & x ∈ ℝ2
Given a point P and a radius R then all the points from space of Real Numbers squared that are distance R from P
However a problem: the set of points that form a circle is not actually a Circle: it is a set of Points. Points and Circles are completely different objects.
Now let us say that Points and Circles both exist. Perhaps we can express a Circle as the definition above.
So Quine is saying that both the set of points C and the expression that defines those points C' are both existing things.
For example these 4 points fit the description of a circle in that they all lies the same distance from the central point. So they are a subset of a C. But they are also a subset of a Square and an infinite number of definitions.
So the collection of points and the definitions that they belong to are indeed different. Looking at this picture you can see four points and then think to fit them into a Square or Circle or any shape you like at will. The shape they fit into is different from the points. Now Quine is saying both the points and the definitions exist.
Now both the definitions of Circle and Square belong to another set "Regular Shapes." And all 3 belong to "Shapes" and then you have powersets and the order of the set of definitions is Aleph 1. So Quine is saying that there are uncountably many existent things.
Okay no problems here yet... TBC
===
So if Quine is going to start arguing that Abstract Mathematical Objects "exist" he is saying that "ideas" exist. Can we agree in this syllogism:
1. All Abstract Mathematical Objects are thinkable.
2. All thinkable things are ideas.
3. Therefore All Abstract Mathematical Objects are Ideas.
>Quine is arguing that Abstract Mathematical Objects "exist" and exist independent of ideas. Like a Unicorn is an idea, but they don't exist. And presumably galaxies existed before the idea. They are not the same. Abstract Mathematical Objects
It is interesting to consider proofs by contradiction gained by assuming that something does not exist. So we can show that something must exist before actually finding it. Constructivists would argue this was impossible. One must assume then that Quine was such a thinker, as he seems to suggest that Mathematical Objects MUST exist before discovery.
Famous philosophical arguments like the Ontological Arguments for God are like this. There must be a "greatest entity." Firstly there must be a greatest entity of mind. And then an entity that does exist is greater than one that does not exist so the greatest entity of mind exists.
>Let me recap exactly what Quine says. Scientific theories imply existence of certain things. Since ME are among the certain things then we must add ME to our ontology.
What do we mean by existence? Let me go back to real basics for my own recollection...
The Andromeda galaxy exists. You can see it, and record data from it. But do "galaxies" exists?
Suppose there are only 2 galaxies: a, b.
That means there is a set G = {a, b}.
G = ∀x | x is a galaxy
There may be another galaxy 'c' that has not been found yet. So we can never know how big the set G is, although we know it is at most countably infinite.
Now one question is: does 'c' exist before it is found?
So we separate the issue into Ontological (what Exists) and Epistemological (what is Known).
The thing to note is that the question is Epistemological. We are asking:
"do we know that c exists, before it is found."
But we can:
"only know something exists by finding it".
So we cannot answer this question unless there is a way to know something without finding it.
One way is by collecting evidence. So we have Bayesian Inference to help us here. But the problem here is that by Bayesian Inference you never have a certainty. As evidence mounts we become more certain but never certain.
This is almost the question of Evolution of Species. Challengers of Evolution look at the difference between variation and species. All evidence for actual evolution has been for gradual changes in form. While people look at Elephants and Giraffes and say how can that difference be accounted for by accumulated variation. They are absolutely "different" species. This idea comes from the Biblical idea of "God make the wild animals according to their kind" [Genesis 1:25]. But a philosopher might say this is a limitation of language. Famously it was Gilbert White who separated the Chiffchaff from the Willow Warbler. To a modern birder two species so different that it's amazing they were confused. Their songs being completely unalike for a start.
As well as the idea of species being cast in stone, it is just as common today to see all life as connected in a family tree. In fact all life was connected before because it was all made by God. You could argue the modern idea of completely separate species could be a "rigid" US view like I'm accusing Quine.
So did the Chiffchaff and Willow Warbler and in fact Wood Warbler exist before Gilbert White identified them as? It's a mixed up question. People saw all these birds but classified them probably under some name like LBJ (little brown job). So all White did was reclassify them. That is give them new names.
LBJ = ∀x | x is a small brown bird.
After Gilbert White's work:
GW = ∀x | x ∈ LBJ & (x is Willow Warbler || x is Chiffchaff || x is Wood Warbler)
So the x are all the same before or after Gilbert White, just the categories different. We can call this Type C discovery.
Russel and Whitehead originally came up with Predicate Logic to solve this ontological problem:
"Unicorns are white."
the problem is what thing are we saying is white? Normally we apply a predicate like The Sun is yellow. This presupposes a Sun and then says it is yellow. No problem. But to say Unicorns are white suggests that there are unicorns. If we say it is false then are we saying that unicorns are not white? So we presume there is a "World of Things", and then ask to select those things that are unicorns and those which are white. We can then make a statement saying there are no such things. True there may be black unicorns but that is a separate statement, we are not committed to unicorns in this sentence.
This "World of Things" is the Ontology and we presume it exists. Type B discovery would be finding an 'x' which fits an existing classification. Like the Higgs Boson. Higgs proposed such an elementary particle before it was found.
∃x | x is Higgs Boson
This was unknown before experiments found an 'x' which fitted Higgs description.
The other possibility is finding an 'x' that does not fit existing descriptions like Dark Matter. There is evidence for something but no one has any idea what it is. This is classic Type A discovery.
This interaction between x and the "bins" we put the x in can lead to all kinds of issues.
Originally there were 2 entities in the sky: the morning star called Phosphorus and the evening star called Hesperus. They had very clear descriptions: one appearing in the morning sky and one in the evening sky. But there was a clue: they never occurred at the same time so we could not be sure they were separate entities. Indeed it turned out that a new theory was possible where they were the same x but due to its orbit and the Earths orbit it sometimes appeared in the evening and sometimes in the morning. This is Type C where a reclassification which reduced the bins, but that included a reduction in the number of 'x' too. Occam's Razor is don't have any unfilled bins. If Phosphorus and Hesperus are exclusively empty then replace them with the one bin Venus with one entity in all the time.
Now where does this get us with Quine. When Quine says that scientific theories include mathematical objects then some 'x' are mathematical objects.
TBC
But then is not discovery just the same as reclassifying it?
The problem with this question is that we can't answer it. If we could answer it then it would be found!
So we can expand Quine's musings to do "Ideas Exist?"
Let us propose the following proposition and syllogism:
1. All Ideas exist
2. The idea of "exist" is an idea.
3. Therefore the idea of "exist" itself exists.
Quine should be getting uneasy now: a use and mention! And from SRH (see blog) we know there is a contradiction lurking in here. In fact if SRH was proven we could just end the proof here. But it isn't so I need to find the actual contradiction.
And I don't have time right now...
But what we are going to see is that a "particular thought" like "oh I just realised I need to go to the gym today because tomorrow we're going shopping" is embedded in time. You have the thought and then you don't have the thought. This is what we mean by the "thought existing."
IF you then try and argue that the idea of the "gym" exists outside this context, like its sitting there in a Platonic Space waiting to be thought by someone then you have 2 types of existence. An actual instance of the thought AND the general abstract entity commonly called Type and Token.
However, and not sure this has been shown before, but once you have made that split you can go mapping isomorphs and splitting recursively forever like Cantor's infinite ascent of numbers. That is to say like a fractal, once you can find resemblance or mapping between the whole and a part created by a split (or any operation), then you can apply the split again on that part and then continue forever recursively.
Types need be split into an ascending hierarchy of Types. Which is interesting because this is Russel and Whiteheads solution to their paradox in Principia Mathematica. However other logics have got around this without types.
Anyway suffice for now to conclude that Quine needs introduce at least 2 types of existence which he does not. The actual instance of a mathematical object in an actual debate or real-time event, and the Abstract existence he is proposing. Obviously Platonic thinking does this but reduces the real-time existence to a "shadow" of the Form thus trying to reduce existence to just a single concern. In that there are many apples which gain their "similarity" through resemblance to a "Normative" Apple in the world of Forms. Biology does this in real life with the type designation. But unlike Plato's mysterious idea world of Forms the "type" specimen is just a particular specimen itself. Plato's "existence of Forms" leads to the exact same contradiction hinted at here because to explain existence you need create new existences (namely forms) and then you end up in an infinite regress.
So broadly isn't Quine doing the same? Anyway definitely no more time now, was opening wiki for actual work not a huge diversion... to be continued...