Isn't this article conflating our formalism of a given abstract entity (like real numbers or integers) with the abstract entity itself? Surely quantities existed long before humans (e.g. there was a quantity of stars in the Milky Way 1 million years ago). And surely ordinals existed long before humans (e.g. there was a most massive star in the Milky Way 1 million years ago).
The article's claim seems to be about the mathematical formalisms humans have invented for integers and real numbers. And I agree that our formalism of integers is simpler and more elegant than our formalism of real numbers. But that could just be because we've done a worse job formalizing real numbers!
> Isn't this article conflating our formalism of a given abstract entity (like real numbers or integers) with the abstract entity itself?
It is arguing that the integers separate from the reals is the formalism and that the abstract entity is the reals.
We also have a formalism of the reals, but it is closer to the abstract entity.
> And I agree that our formalism of integers is simpler and more elegant than our formalism of real numbers. But that could just be because we've done a worse job formalizing real numbers!
As we create better more useful formalisms, we interact more with the formalism than the thing itself. It is like putting on oven mitts to pick up a hot tray from the oven. This has already happened with the reals!
Consider the question of if the reals are well-ordered or not. The question applies to the actual entity of the reals itself, but the absoluteness theorem shows that the question has no arithmetic consequences. You can simple ignore the question. Thus, those seeking the mental comfort and utility of the formalism do not have to concern themselves with the true nature of the real number line.
> And I agree that our formalism of integers is simpler and more elegant than our formalism of real numbers. But that could just be because we've done a worse job formalizing real numbers!
Everything you can express in integers you can express in reals, but there are many things expressable in reals not possible in integers. It would be surprising if the formalism for a thing that completely supersets another thing had an equally simple formalism
I'm not suggesting the two formalisms should be equally simple. But surely it's not controversial to claim that formalizing the reals involves much more advanced mathematics (and runs into much deeper problems) than formalizing the integers. I'd argue that this disparity is slightly surprising, given that both the integers and the reals are ubiquitous in essentially all branches and levels of mathematics.
Actually in math it's very common for the more general system to be simpler. Compare for example the prime numbers with the integers, or general groups with finite simple groups and the monster group.
I mean, this is one of the deepest questions of philosophy. Do or can concepts and categories exist without the beings that create them?
Does tuna casserole exist independently of humans? If not, how is the idea of the number 7 different from the idea of a tuna casserole? Or what about the concept of decision by majority, which isn't as basic as 7, but doesn't have the physicality of a casserole?
Modern science is derived from Christian Scholasticism from the middle ages so this way of talking and thinking about science as being divinely originated is only unusual in the past couple centuries or so.
It is from that era that they developed systems of rigorous debate, formal logic, and things like peered reviewed papers that we call "the scientific method".
As far as the history of these sorts of mathematical discussions the concept of negative numbers didn't exist until the 15 century. I am sure that each new concept was faced with some resistance and debate on its true nature before it became widely accepted.
So I am sure that somebody looking through the historical record could find all sorts of wild quotes from different theologians trying to grasp new concepts and reconcile them with existing mathmatical standards.
The double entendre of "having studied Agrippa" in the footnotes is probably going to go unnoticed unless someone mentions it here. Contemporary to the cited Camillo Agrippa, fencing master, was Henrichus Cornelius Agrippa, whose collections on philosophy and occultism are much more relevant to the topic. H.C. Agrippa's work is still considered authoritative in its fields: the numbers represent the ideas, which were both created in the first moments by God. the difference between set of reals and set of integers might have a correlation to the difference between the set of all expressible concepts, and (the smaller) set of actually meaningful concepts. Maybe some computability theory could be tossed in there too.
I was thinking of making a "the other Agrippa" reference since with half my friends if you say Agrippa will think Camillo and the other half will think Henrichus Cornelius. I've only read Camillo Agrippa, I've been meaning to read Henrichus. Is he worth reading? What do you recommend?
> H.C. Agrippa's work is still considered authoritative in its fields: the numbers represent the ideas, which were both created in the first moments by God.
That does sound like the computability idea of numbers being programs.
```
In the beginning was the Word, and the Word was with God, and the Word was God.
The same was in the beginning with God.
All things were made by him; and without him was not any thing made that was made.
```
The "Word" in Greek is translated from logos which also means logic, rational, metric.
```
In the beginning was the Number, and the Number was with God, and the Number was God.
```
The reals aren't algebraically under multiplication; a simple equation like x*x=1 can't be handled in real numbers. The complex numbers are algebraically closed. So I suspect that God created the complex numbers.
God certainly had a fondness for the real subset. Measurements are real scalars -- so much so that it really does look like God created the reals. That's what's important to us. But the fundamental laws seem to require the complex numbers (or their equivalent, like matrices), and closure under arithmetic operations really does feel like it should be a requirement for the reality of the universe.
I'm more troubled by the fact that almost all real numbers are uncomputable (same goes for complex numbers, of course). It's very straightforward to see that this is the case, but the mathematics involved to even begin to ponder questions like "under which operations is the set of computable reals not closed" seem to be far over my head.
I infer from footnote 10 that an unspoken subtext of this is that footnote 1 is that while the reader may choose a (simplistic) atheist's formulation of the idea, the author does not,
which would be consistent with their interest in the question of the "divine" and human reasoning at all, especially as argued about by theologically inclined philosophers much admired by Judeochristians.
That subtext being, discovering that our models or knowledge are incomplete somehow increases the territorty of what he's calling mysterious. By which I take it he means, knowable to and to not beat around the burning bush, attributable to the divine. By which I take it for him that he means a Judeochristian god.
One of the great and persistent bemusements of my adulthood is discovering that other adults take their religiosity not just seriously but central to their understanding of themselves, and their context generally.
It's a relief that such people have participated in construction of a society within which such beliefs are considered personal, as it saves a lot of embarassment for people such as myself, who find such notions wince-inducing, and, both their origins and utility quite transparent.
> I infer from footnote 10 that an unspoken subtext of this is that footnote 1 is that while the reader may choose a (simplistic) atheist's formulation of the idea, the author does not
Author here. I'd describe myself as atheist/agnostic.
I just dislike the God of the gaps argument. I understand its utility in debating the dishonest moving goal post arguments of young earth creationists, but taken outside of that debate, I don't think it holds water.
> One of the great and persistent bemusements of my adulthood is discovering that other adults take their religiosity not just seriously but central to their understanding of themselves, and their context generally.
I take religion and religious questions seriously. If I took religion less seriously I'd be religious because I enjoy religion. We owe each other a certain level of honesty on truly serious matters even if it is uncomfortable.
> By which I take it he means, knowable to and to not beat around the burning bush, attributable to the divine. By which I take it for him that he means a Judeochristian god.
That is certainly how a 16th Century religious Italian would understand it and I find that an interesting perspective to contrast with my own somewhat blander late-modernist beliefs. One of the reasons I enjoy reading books from prior ages is seeing how much that was taken for granted as a universal truth of that culture has changed.
Since quantum uncertainty (in a finite universe) basically says that you can't measure anything with infinite precision - I'd argue that God created, at most, the Rational Numbers. The Reals might be the closure ( https://en.wikipedia.org/wiki/Closure_(topology) ) of the Rationals - but doing that was the work of man.
Isn't this article conflating our formalism of a given abstract entity (like real numbers or integers) with the abstract entity itself? Surely quantities existed long before humans (e.g. there was a quantity of stars in the Milky Way 1 million years ago). And surely ordinals existed long before humans (e.g. there was a most massive star in the Milky Way 1 million years ago).
The article's claim seems to be about the mathematical formalisms humans have invented for integers and real numbers. And I agree that our formalism of integers is simpler and more elegant than our formalism of real numbers. But that could just be because we've done a worse job formalizing real numbers!
> Isn't this article conflating our formalism of a given abstract entity (like real numbers or integers) with the abstract entity itself?
It is arguing that the integers separate from the reals is the formalism and that the abstract entity is the reals.
We also have a formalism of the reals, but it is closer to the abstract entity.
> And I agree that our formalism of integers is simpler and more elegant than our formalism of real numbers. But that could just be because we've done a worse job formalizing real numbers!
As we create better more useful formalisms, we interact more with the formalism than the thing itself. It is like putting on oven mitts to pick up a hot tray from the oven. This has already happened with the reals!
Consider the question of if the reals are well-ordered or not. The question applies to the actual entity of the reals itself, but the absoluteness theorem shows that the question has no arithmetic consequences. You can simple ignore the question. Thus, those seeking the mental comfort and utility of the formalism do not have to concern themselves with the true nature of the real number line.
> And I agree that our formalism of integers is simpler and more elegant than our formalism of real numbers. But that could just be because we've done a worse job formalizing real numbers!
Everything you can express in integers you can express in reals, but there are many things expressable in reals not possible in integers. It would be surprising if the formalism for a thing that completely supersets another thing had an equally simple formalism
I'm not suggesting the two formalisms should be equally simple. But surely it's not controversial to claim that formalizing the reals involves much more advanced mathematics (and runs into much deeper problems) than formalizing the integers. I'd argue that this disparity is slightly surprising, given that both the integers and the reals are ubiquitous in essentially all branches and levels of mathematics.
Actually in math it's very common for the more general system to be simpler. Compare for example the prime numbers with the integers, or general groups with finite simple groups and the monster group.
I mean, this is one of the deepest questions of philosophy. Do or can concepts and categories exist without the beings that create them?
Does tuna casserole exist independently of humans? If not, how is the idea of the number 7 different from the idea of a tuna casserole? Or what about the concept of decision by majority, which isn't as basic as 7, but doesn't have the physicality of a casserole?
Modern science is derived from Christian Scholasticism from the middle ages so this way of talking and thinking about science as being divinely originated is only unusual in the past couple centuries or so.
It is from that era that they developed systems of rigorous debate, formal logic, and things like peered reviewed papers that we call "the scientific method".
As far as the history of these sorts of mathematical discussions the concept of negative numbers didn't exist until the 15 century. I am sure that each new concept was faced with some resistance and debate on its true nature before it became widely accepted.
So I am sure that somebody looking through the historical record could find all sorts of wild quotes from different theologians trying to grasp new concepts and reconcile them with existing mathmatical standards.
The double entendre of "having studied Agrippa" in the footnotes is probably going to go unnoticed unless someone mentions it here. Contemporary to the cited Camillo Agrippa, fencing master, was Henrichus Cornelius Agrippa, whose collections on philosophy and occultism are much more relevant to the topic. H.C. Agrippa's work is still considered authoritative in its fields: the numbers represent the ideas, which were both created in the first moments by God. the difference between set of reals and set of integers might have a correlation to the difference between the set of all expressible concepts, and (the smaller) set of actually meaningful concepts. Maybe some computability theory could be tossed in there too.
I was thinking of making a "the other Agrippa" reference since with half my friends if you say Agrippa will think Camillo and the other half will think Henrichus Cornelius. I've only read Camillo Agrippa, I've been meaning to read Henrichus. Is he worth reading? What do you recommend?
> H.C. Agrippa's work is still considered authoritative in its fields: the numbers represent the ideas, which were both created in the first moments by God.
That does sound like the computability idea of numbers being programs.
``` In the beginning was the Word, and the Word was with God, and the Word was God. The same was in the beginning with God. All things were made by him; and without him was not any thing made that was made. ```
The "Word" in Greek is translated from logos which also means logic, rational, metric.
``` In the beginning was the Number, and the Number was with God, and the Number was God. ```
Looking around on the internet, there is a lot here: https://en.wikipedia.org/wiki/Rationes_seminales
The reals aren't algebraically under multiplication; a simple equation like x*x=1 can't be handled in real numbers. The complex numbers are algebraically closed. So I suspect that God created the complex numbers.
God certainly had a fondness for the real subset. Measurements are real scalars -- so much so that it really does look like God created the reals. That's what's important to us. But the fundamental laws seem to require the complex numbers (or their equivalent, like matrices), and closure under arithmetic operations really does feel like it should be a requirement for the reality of the universe.
I'm more troubled by the fact that almost all real numbers are uncomputable (same goes for complex numbers, of course). It's very straightforward to see that this is the case, but the mathematics involved to even begin to ponder questions like "under which operations is the set of computable reals not closed" seem to be far over my head.
I think you mean x*x = -1, for which I agree with your point.
I infer from footnote 10 that an unspoken subtext of this is that footnote 1 is that while the reader may choose a (simplistic) atheist's formulation of the idea, the author does not,
which would be consistent with their interest in the question of the "divine" and human reasoning at all, especially as argued about by theologically inclined philosophers much admired by Judeochristians.
That subtext being, discovering that our models or knowledge are incomplete somehow increases the territorty of what he's calling mysterious. By which I take it he means, knowable to and to not beat around the burning bush, attributable to the divine. By which I take it for him that he means a Judeochristian god.
One of the great and persistent bemusements of my adulthood is discovering that other adults take their religiosity not just seriously but central to their understanding of themselves, and their context generally.
It's a relief that such people have participated in construction of a society within which such beliefs are considered personal, as it saves a lot of embarassment for people such as myself, who find such notions wince-inducing, and, both their origins and utility quite transparent.
> I infer from footnote 10 that an unspoken subtext of this is that footnote 1 is that while the reader may choose a (simplistic) atheist's formulation of the idea, the author does not
Author here. I'd describe myself as atheist/agnostic.
I just dislike the God of the gaps argument. I understand its utility in debating the dishonest moving goal post arguments of young earth creationists, but taken outside of that debate, I don't think it holds water.
> One of the great and persistent bemusements of my adulthood is discovering that other adults take their religiosity not just seriously but central to their understanding of themselves, and their context generally.
I take religion and religious questions seriously. If I took religion less seriously I'd be religious because I enjoy religion. We owe each other a certain level of honesty on truly serious matters even if it is uncomfortable.
> By which I take it he means, knowable to and to not beat around the burning bush, attributable to the divine. By which I take it for him that he means a Judeochristian god.
That is certainly how a 16th Century religious Italian would understand it and I find that an interesting perspective to contrast with my own somewhat blander late-modernist beliefs. One of the reasons I enjoy reading books from prior ages is seeing how much that was taken for granted as a universal truth of that culture has changed.
Since quantum uncertainty (in a finite universe) basically says that you can't measure anything with infinite precision - I'd argue that God created, at most, the Rational Numbers. The Reals might be the closure ( https://en.wikipedia.org/wiki/Closure_(topology) ) of the Rationals - but doing that was the work of man.