« first day (7 days earlier)      last day (733 days later) » 

user21820
user21820
06:08
@user170039: I didn't want to give specific names in my reply to you on Main, but here is one: Asaf Karagila. There are many others, but Asaf has made his views quite clear in various places, even if not frequently.
From his notes:
> Long before them, we like to imagine, men counted mammoths, or rocks, or tribe members, and at some point someone realized that quantity is abstract. So the quantity obtained by adjoining two apples to another apple is the same as the number of three children. From there, we like to think, arithmetic developed, and as society became more complex the need for better abstractions grew and fractions, zero, irrational numbers and many other concepts became commonplace.
This is pretty much the same as what I said about basic arithmetic; it abstracts basic counting in the real world.
> [...] In order to count how many apples you have, first you need to identify that there is a collection of apples to count. In order to count the members of your tribe, first you need to identify that there is a collection “of people in your tribe”. We see, if so, that the notion of a collection is as fundamental than the idea of counting. [...]
I agree with this cited portion, but not completely with the omitted parts. The reason is that it is absolutely correct when we talk about collections as types. Not as entities that always can tell you whether an object is a member or not.
> So what are sets? How do we know what properties they have? Well, we postulate a list of axioms which seem self-evident from our understanding of collections in real life, and finite collections we can “test by hand”.
This next sentence clearly validates my assessment, because Asaf is endorsing jumping from the finite to the general (possibly infinite), which implicitly endorses basic arithmetic but fails to provide justification that the infinite behaves the same way.
> Over the course of our lives we develop intuition that allows us to say whether or not something is true or false. This intuition is usually developed from our physical experience, and the way we reason internally to understand the things we see. Mathematics in general, and set theory in particular do not deal directly with reality, but rather with well-defined abstract objects. In set theory the objects of interest are arbitrary sets which represent nothing physical.
user21820
user21820
06:25
This comment clearly shows that Asaf believes that physical experience is what we use to verify meaningfulness of mathematics that is physically relevant. It also clearly shows that Asaf does not believe that more modern mathematics deals with only physically meaningful entities, when he says that some arbitrary sets represent nothing physical!
He said the same thing a few times on Math SE in the past but I don't know how to find all of them. Here is another strikingly clear example:
9
A: How Do You Know If Mathematical Definition Matches Up With Reality?

Asaf KaragilaThe point of mathematics is not to match to reality. This might have been the one of the origins of mathematics, but it has grown and evolved long beyond that point. Mathematics is its own universe, and it deals with assumptions and consequences. How do you know that mathematical definitions ma...

If it's not clear enough, Asaf states not only that (in his view) mathematics is not intended to match reality (and if so then in my view not philosophically sound), but intriguingly also claims that in reality there is no infinite object/set, nothing larger than 2^(10^11), no fractions or irrationals, no negative numbers, but there is 1,2,3,... up to some n < 2^(10^11). This claim clearly mirrors my claim that PA is at least at small scales a good approximation of some entity in reality!
@user170039: Lest you think that Asaf is an outlier, let me tell you that many logicians subscribe to a similar viewpoint, if for no better reason than what I explained about finite strings underlying logic itself. Here's another example:
6
A: Usage of set theory in undergraduate studies

Monroe EskewHistorically, mathematics did not begin with set theory, and most working mathematicians today do not really care about foundations. Furthermore, it is known that not much axiomatic strength is actually needed for undergraduate math: The body of research in reverse mathematics has establishe...

Monroe goes even further than Asaf and boldly claims the "unquestionable reality of basic mathematics" as being eternal, whereas "Set Theory and Category Theory are ideologies that may not withstand the test of time". Note that in a comment he further says that "they are somewhat removed from the absolute math for which they serve as foundation".
If that wasn't clear enough, he had this to say: "if you know anything about relative consistency results, or even just some history of math, it is clear that basic math is much more eternal than current foundations, which may actually die".
user131753
07:32
I will reply to the following comments of yours,
user131753
@user170039: If you believe that logic itself is meaningful, you have to believe the basic properties of finite strings. That gives essentially a real-world model of PA−. From that model of PA− you get a model of PA from the standard numerals. I have definitely told you this numerous times before; do read and understand what I tell you to before judging my statements without a background in logic. — user21820 2 hours ago
user131753
and,
user131753
@user170039: By my above comment every logician who believes their work is meaningful must also believe that PA is (at least at small scales) meaningful. So pick any logician who believes that ZFC has nothing to do with the real world. A separate and perhaps more compelling argument is that you use 2048-bit RSA decryption every time you visit Math SE. If you seriously doubt that PA (or at least the fragment needed to prove that RSA decryption works) has real-world meaning, then you have a serious inability to explain why you can use Math SE. — user21820 2 hours ago
user131753
Regarding the first comment, I don't understand what you mean by being logic itself to be meaningful because according to one of your previous comments if you hold that "A sufficient notion for my purposes is that it has an interpretation in the real world..." then I have some questions,
user131753
1. What is a real world interpretation of logic?
user131753
07:45
2. Why are there different systems of logic?
user131753
07:55
3. Suppose for the time being I believe that logic itself is meaningful. Then according to you I have to believe the basic properties of finite strings. For the time being let us suppose that too. But how exactly then it follows that "gives essentially a real-world model of PA−. From that model of PA− you get a model of PA from the standard numerals" is not clear to me.
user131753
No doubt (for me at least) there is an informal and/or intuitive connection between a model of PA− and the so called "real world" but without a precise formulation of the concept of real world it doesn't make much sense to talk about a real-world model of PA−.
user131753
Regarding the second comment, when I talked with one of my professor regarding this issue (whose philosophical position can be said to be a sort of extreme game formalism), he simply said, "It's a conicidence!"
user131753
08:12
→ 1 message moved to trash
user131753
→ 1 message moved to trash
user131753
I have removed one of your messages @user21820 as it was not particularly relevant to the ongoing discussion.

« first day (7 days earlier)      last day (733 days later) »