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!
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.
Why believe in abstract entities at all, as something distinct from the formalism? We have various formal or abstract concepts, that are useful in science and its applications, but they make contact with reality only through these uses: the natural numbers are used in counting, which is useful because there are many physical objects or events that are similar enough to each other that their differences can be neglected, then it's meaningful to count them and come up with a number for the collection, e.g. apples in a basket, planets in the solar system, or whatever.
Real numbers developed from our perception of continuous physical magnitudes, and from the usefulness of applying the concept of number to these magnitudes. Then the formalism of the real numbers was developed based on mathematical constructions from the natural numbers/integers.
We don't have to posit some abstract entity that the concept of real numbers refers to: it's a symbolic or mathematical construction that helps us in reasoning about (what we perceive as) continuous magnitudes, which aren't abstract, but concrete aspects of our experience of the world
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
Are you sure there is anything we can express in the Reals that isn't an integer in disguise?
The first answer might be the sqrt(2) or pi, but we can write a finite program to spit out digits of those forever (assuming a Turing machine with integer positions on a countably infinite length tape). The binary encoding of the program represents the number, and it only needs to be finite, not even an integer at infinity.
Then you might say Chaitin's constant, but that's just a name for one value we don't know and can't figure out. You can approximate it to some number of digits, but that doesn't seem good enough to express it. You can prove a program can't emit all the digits indefinitely. And even if you could, is giving one Real number a name enough? Names are countable, and again arguably finite.
It seems to me we can prove there are more Reals than there are Integer or Computable numbers, but we can't "express" more than a finite number of those which aren't computable. Integers in disguise.
There is no fundamental unit of "star". Maybe we can talk about electrons or protons or something, but what is and is not a star is a model, not a reality.
Concretely, a bundle of pre-stellar gasses at some point transitions to being a star, but when in that time spectrum does it make that transition? When in the process of stellar exhaustion does it stop being a star?
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.
No, I don't think so. It seems much more based on ancient Greek geometry and logic, the Indian numeral system and Arabic algebra. Modern science really took off after Galileo, at the time when the ancient Greek works were recovered in Europe and could be synthesized with the arithmetic and algebra of these other cultures. Galileo himself credits the "divus" Archimedes as his main inspiration.
What aspects of Christian scholasticism do you think developed into modern science?
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?
No of course not, but that's not the question. The question is whether the concepts were created by the beings.
In a system with 7 stars the number of stars doesn't change when all humans die or humans never even developed.
Which would mean integers are baked in, rationals too, but non-constructible reals (essentially all reals, given any degree of approximation) are a useful abstraction but don't actually exist in any way.
Reals are not real.
(Roughly) Equivalently: There may be no perfect circles in nature.
For one thing, they're speculative: the current theories that give the most precise and accurate predictions within their respective domains of applicability are general relativity and quantum field theory. These theories are based on continuous space and time, and no attempt to base them on discretized space and time has been successful (AFAIU both QFT and GR rely on Lorentz invariance, which means there's no absolute rest frame, hence no absolute unit of time and space, but a discretized spacetime would require absolute unit values for space and time, hence an absolute rest frame).
Should we conclude then that the reals are real, because they're components of our best current physical theories? Maybe, maybe not: these are features of our current best models, but we don't know, and possibly will never know, the ultimate nature of physical space and time.
For another thing, even if space and time are fundamentally discrete, there's still no doubt that the mathematical theories based on real numbers are effective in making predictions, and we would still like to use them. That's means they should have some logical foundation which can guarantee that reasoning using them is correct.
But this claim is nowhere made in the comment? Like clearly the transfinite ordinals aren't real, but no one would say that implies they aren't a very useful mathematical idea (and also just interesting in and of themselves).
Constructible reals are also a subclass of reals, but you can't claim anything about the class of reals, which are vastly dominated by un-constructible reals, because constructible reals have a property.
There are many reasons to doubt un-constructible numbers exist in nature.
Just for starters, you can never actually define a specific un-constructible real. If you did, you would have defined it, making it constructible.
An un-constructible real requires infinite information to define. Not infinite digits (pi is constructible, e is constructible), but an infinite list of uncompressible digits, or some other expression with infinite numbers of symbols!
The name "reals" is highly deceptive/unfortunate. (What could be more reasonable than a "real" number?)
We need a pithier name for constructible numbers, and that is what should be introduced along with algebra, calculus, trig, diff eq, etc.
None of those subjects, or any practical math, ever needed the class of real numbers. The early misleading unnecessary and half-assed introduction of "reals" is an historical educational terminological aberration.
(It would be nice to rename "real" numbers, to mean actually real numbers that we could actually use. But given the generations of confusion that would incur, I propose "actual numbers", to be all constructible numbers. Nobody but mathematicians, who play abstract games with higher order infinities, need "real" 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.
0.59999999999999999999999...
+ 0.00000000000000000000000...
------------------------------
0.?
I think you can make pathological cases for every arithmetic operation, so maybe (I'm not sure) none of the operations are computable. (Need to be careful with the definitions though, and I'm being pretty sloppy)
> 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
> All abstract concepts brought about solely by humans.
That is a defendable philosophical position to hold but many thinkers disagree with it or at least admit there is more nuance [1]. People have been debating the question of the independent reality of mathematical concepts for thousands of years.
If abstract concepts are brought about solely by humans, does that mean the first human to invent a proof of some property actually decides the reality of that property? If Godel hadn't proved incompleteness, could another mathematician created a completeness proof instead? On the hand, if that isn't the case and the truth or falsity of a math statement exists prior a human thinking it, doesn't it have some reality beyond human thought? What about mathematical statements which are true which can't be proven?
[0]: Platonism in the Philosophy of Mathematics https://plato.stanford.edu/entries/platonism-mathematics/
[1]: Mathematics: Discovery or Invention? https://royalinstitutephilosophy.org/article/mathematics-dis...
That said I don't think there's much to be gained from dragging God into it all. Even He would probably have a headache changing pi to 3.2.
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.
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.
For example, in fault tolerant quantum computing, rotations are synthesized using sequences of 45 degree rotations around the X, Y, and Z axes. The matrices that describe those 45 degree rotations contain rational and irrational numbers (in particular: sqrt(2)). If those irrational numbers are actually truncated, this would have observable consequences. You'd need a sufficiently large quantum computer running for sufficiently long to do sufficiently accurate tomography of sequences of those rotations in order to resolve the truncation, and to be frank some of those "sufficientlies" would be very impractical to achieve especially if the truncation was small (and woe unto you if adding more qubits somehow reduces the amount of truncation!), but in principle it'd be possible.