Thanks and sorry for posting a topic which is not a typical discussion — keystone

And refreshing it is for these times. Not a mention of God or Jesus or climate change.

I was a professional mathematician for many years, focusing on complex analysis, teaching and writing a few papers, and still do modest research. And although I accept ideas like the set of real numbers and associated cardinalities, I have never used infinity as anything more than unboundedness. To all intents and purposes my mathematics has been infinity free.

The forum has had a number of discussions about this topic, but that's no reason for you to avoid bringing it up in a new thread or resurrecting an old thread. There are some sharp people here.

Not a mention of God or Jesus or climate change. — jgill

:rofl: Don't be so sure.

To Cantor, his mathematical views were intrinsically linked to their philosophical and theological implications – he identified the Absolute Infinite with God, and he considered his work on transfinite numbers to have been directly communicated to him by God, who had chosen Cantor to reveal them to the world. — Wikipedia

God = $\infty$.

The OP is an atheist, mathematically speaking that is.

God = ∞. — Agent Smith

The tone of the OP does not suggest Cantor's theological nonsense.

The tone of the OP does not suggest Cantor's theological nonsense. — jgill

That's true. Apologies if my comment offended you in any way.

The OP mentions Aristotle's distinction of actual vs. potential infinities. The Wikipedia page on the subject doesn't explain the difference between the two all that well. My own take is that Aristotle assumes that for something to be actual, that thing has to have an "end"; since $\infty$ has no end (it's synonym is endless) it can't be actual. The only alternative then is to "exist" only as pure potential.

Finitism is kinda sorta echoed by Agent Smith in The Matrix $\downarrow$

I have never used infinity as anything more than unboundedness. — jgill

That can't be true. Calculus is all about infinity.

Cantor's theological nonsense — jgill

If you believe that mathematics represents some sort of idealistic ultimate reality, perhaps Cantor's infinities could be seen as "theological." But then, all of math is. Math is all fun and games, that just turns out to work. People used to think that "0" is absurd. And negative numbers. And "i."

I accept ideas like the set of real numbers and associated cardinalities — jgill

The set of real numbers is at the center of my discontent. It puts points as the foundational building block of mathematics and I find it hard to imagine that something (an n-dimensional continuum) can be constructed from nothing (0-dimensional points).

I have never used infinity as anything more than unboundedness. To all intents and purposes my mathematics has been infinity free. — jgill

This is why my concerns are more philosophical than mathematical. I think changing our philosophy (removing points from the foundation) will have little impact on the actual mathematics that we do.

The forum has had a number of discussions about this topic, but that's no reason for you to avoid bringing it up in a new thread or resurrecting an old thread. — jgill

If I can't find a mentor, I might do just that.

The tone of the OP does not suggest Cantor's theological nonsense. — jgill

I do feel that there is a little bit of Cantor's nonsense implied in any view that supports actual infinities. I've heard people say that the paradoxes entwined with actual infinities are beautifully mysterious...I just think they demonstrate the flaws of the concept of actual infinity.

Calculus is all about infinity. — T Clark

I would argue that calculus done right (with limits) is all about potential infinities.

I would argue that calculus done right (with limits) is all about potential infinities. — keystone

All mathematics is about "potential" entities. So what we gonna do? Round pi off to 3.14? 3.14159? How many decimal places do I need to get to the real pi?

I find it hard to imagine — keystone

History shows that is a bad standard by which to judge a concept.

I'll stop now. I'm not a mathematician, just an engineer like you.

All mathematics is about "potential" entities. So what we gonna do? Round pi off to 3.14? 3.14159? How many decimal places do I need to get to the real pi? — T Clark

Why can't we just say that pi is not a number? Instead, it is an algorithm (e.g. pick your favorite infinite series for pi) used to generate a number. This algorithm is potentially infinite in that we can never complete it, but we can certainly interrupt it to generate a rational number. If you interrupt it, maybe you'll get 3.14. Actual infinity only comes into play if you claim that the algorithm can be completed, in which case it would generate a real number - a number with actually infinite digits. This is what I would like to challenge.

History shows that is a bad standard by which to judge a concept. — T Clark

Perhaps I should have written that I believe it is impossible to imagine assembling points to form a continuum. A bit of magic is needed to make the leap from a finite collection of points forming nothing to an infinite collection of points forming a continuum.

Perhaps I should have written that I believe it is impossible to imagine assembling points to form a continuum. A bit of magic is needed to make the leap from a finite collection of points forming nothing to an infinite collection of points forming a continuum. — keystone

That makes the real numbers a challenging and intriguing subject.

I have never used infinity as anything more than unboundedness. — jgill

That can't be true. Calculus is all about infinity. — T Clark

$x\to \infty$ means x gets larger without bounds. Limits

That makes the real numbers a challenging and intriguing subject. — jgill

Maybe not as challenging as you think. I think the application of real numbers would remain largely unchanged, it just requires a reinterpretation. If convention says that in between the rational points lies irrational points, I would say that in between the rational points lies continua (e.g. little lines). With this reinterpretation,

• Proofs the irrational numbers/points exist could be reinterpreted as proofs that continua exist.
• Proof that length comes from irrational numbers/points could be reinterpreted as proof that length comes from lines.

There's a simplicity and intuitive appeal to this reinterpretation. What challenge do you see?

That makes the real numbers a challenging and intriguing subject. — jgill

Maybe not as challenging as you think. — keystone

I was speaking of currently accepted set theory, not challenges of it.

I was speaking of currently accepted set theory, not challenges of it. — jgill

Oh sorry, I see what you're saying. I think I have a grasp of how real numbers play into accepted set theory but it is challenging for me to envision the existence of a set of all natural numbers. Without assuming its existence, accepted set theory doesn't get far off the ground.

Why can't we just say that pi is not a number? Instead, it is an algorithm (e.g. pick your favorite infinite series for pi) used to generate a number. This algorithm is potentially infinite in that we can never complete it, but we can certainly interrupt it to generate a rational number. If you interrupt it, maybe you'll get 3.14. Actual infinity only comes into play if you claim that the algorithm can be completed, in which case it would generate a real number - a number with actually infinite digits. This is what I would like to challenge. — keystone

A limitation of that conceptualisation, is that it asserts what might be considered an unnecessarily rigid ontological distinction between functions (intension) and data (extension), which is surely a matter of perspective, i.e the language one uses. Also, recall incommensurability; the length of diagonal lines in relation to a square grid have a length proportional to sqrt(2). The decimal places of sqrt(2) are only "infinite" relative to the grid coordinates.

Any computable total function N --> N can be regarded as a number, whose value is equal to the potentially infinite sequence of outputs it encodes. e.g '3' can be identified with the constant function f(n) = 3, whilst pi can be identified with the computable function whose values if executed are the potentially infinite sequence g(0) = 3, g(1) = 3.1, g(2) = 3.14 ... These numbers can be compared positionwise, with arithmetical operations defined accordingly. However, there are only a subcountable number of such functions, meaning that any set that contains some of these functions either doesn't contain all of them,or contains errors i.e partial functions that fail to halt on certain inputs, to recall the halting problem.

That said, it could be argued that the concept of exact and correct computation, whereby a computer program or function specification is translated by man or machine to a precise and correct result of execution, is an ideal platonistic notion that is incompatible with the austere epistemic and metaphysical conservatism of finitism. In which case one wants a purely extensional treatment of mathematics that doesn't appeal to any notion of computation, in which case see Brouwers intuitionism for a calculus built around choice sequences that appeal only to the existence of resources for memorising data generated by a creating subject.

Perhaps I should have written that I believe it is impossible to imagine assembling points to form a continuum. — keystone

Like many who are philosophically inclined, I am happy to accept actual infinities as a useful mathematical simplification – an epistemic trick – but not something that makes proper ontological sense.

But note also that simply switching your ontological support from parts to wholes – from 0D points to 1D lines – doesn't fix the deeper issues. You just set yourself up for the same puzzle at the next geometric level – where we have to glue together the infinity of 1D lines that construct the 2D plane, the infinity of 2D planes that make up the 3D volume, the 4D hyperspace, and so on, presumably all the way up until we hit ∞D space.

So the idea of 0D points – some kind of absolute notion of discreteness – is offensive to the ontic intuition. But the same should apply to its dichotomous "other", the idea of an absolute continuity as the alternative.

We need a more subtle metaphysics. We need an intuition that itself sees parts and wholes, the discrete and the continuous, as the two emergent parts of the one common rational operation.

This is simple enough. It is just the unity of opposites of ancient times, the dialectics of Hegel, the semiotics of Peirce (in particular).

This would see the discrete and the continuous as being each others limiting case. Each stands as the measure of its other. The two form an inverse or reciprocal relation. A dichotomy is that which is both mutually exclusive and jointly exhaustive. You boil everything down to two precisely opposed limitations. And they are the measure of each other in the form that the discrete = 1/continuity, and continuity is similarly = 1/discreteness.

What does this mean for number lines? It says that while we must think of the 1D whole being constructed of 0D points, that claim must be logically yoked to its "other" of each 0D point existing to the degree the 1D continuity of the line has in fact been constrained.

Constraint is the "other" of construction. Global constraint is the downward causality of holism that matches the notion of construction, which is the upwards causality of reductionism.

So the number line is composed of its 0D parts to the degree that it is also able to limit the continuity that would deny the presence of such 0D parts. And we see this in the emergence of the matching limits of the infinite and the infinitesimal.

Infinite = 1/infinitesimal, and infinitesimal = 1/infinite. Each value relies on its "other" as the source of its own crisp identity. The number line – as our little dichotomised universe of possibility – has to be able to express both polar extremes to express either polar extreme. The points of the line are only as unboundedly tiny as the unbounded extent of the line can ensure this to be the case.

Step back and this fits the kind of metaphysics where what is real emerges as a limitation on an Apeiron or Vagueness – an unbounded potential or limitless everythingness.

The number line would start off as any and every kind of possibility. The principle of noncontradiction would fail to apply to any statement about it, and so it would be formally "vague".

But then a reciprocal operation gets going. You have a separation in which a kind of locality starts to appear within a kind of globality. You get a move towards a collection of parts that simultaneously - in equal reciprocal measure – reveals the larger context that is the continuous whole.

I mean it doesn't even make sense to talk about 0D points except in the context, or in contrast, with the presence of the 1D line, right?

And the more you dig down and keep finding ever more definite points – the infinitesimals – the stronger and better defined becomes the global continuum that has the kind of structure in which such a set of points could clearly exist. You get the co-arising of the continuum. And indeed, the whole Cantorian kit and kaboodle if you keep adding richer structure to your story.

Why not? The more you develop the definition of "the discrete", the more that is based on the refinement of the definition of "the continuous". The two directions of determination are reciprocally yoked. And once you see that, the whole deal becomes less ontologically contentious.

Although it also becomes more ontologically contentious in that you have to shift folk from thinking that the discrete vs continuous issue is an either/or question, to realising that it is a dialectical one. Both co-arise from the common ground of a logical vagueness. Hence without both, you ain't got either.

A limitation of that conceptualisation, is that it asserts what might be considered an unnecessarily rigid ontological distinction between functions (intension) and data (extension), which is surely a matter of perspective, i.e the language one uses. — sime

Can you explain this to me from a computer programming perspective? In your comparison, is the data the output of the function? A function can return a function, but it can also return another object type, like a string. In the latter case, there is a type distinction between between the function and its output, but I don't see how this is unnecessarily rigid. I suspect I'm missing your point.

Also, recall incommensurability; the length of diagonal lines in relation to square grid have a length proportional to sqrt(2). The decimal points of sqrt(2) are only "infinite" relative to the grid coordinates. — sime

Can you conceive of a computer that can display a line of exact length sqrt(2)? In reality something will give, whether it's the gridlines or the diagonal, some imperfection (or uncertainty) will be inserted into the system to allow everything drawn to be 'rationally'. Might the abstract world face the same constraints? If we are certain that the grid lines are perfect, why can't we just claim that we are uncertain of the actual length of the diagonal and instead label it with a potentially infinite algorithm for calculating the length (corresponding to sqrt(2))? In the end, the math is the same, but the philosophy is different since I'm not assuming that sqrt(2) is a number and I'm not assuming that unending processes can be completed.

That said, it could be argued that the concept of exact and correct computation, whereby a computer program or function specification is translated by man or machine to a precise and correct result of execution, is an ideal platonistic notion that is incompatible with the austere epistemic and metaphysical conservatism of finitism. — sime

Yes, I agree to some extent. As mentioned above, if we move away from actual infinity I think we need to allow some uncertainty to creep into our systems. But what's so bad about that? Why do we need our computers to complete an endless computation of sqrt(2)? When we work with sqrt(2) we never actually work with the decimal expansion.

from 0D points to 1D lines – doesn't fix the deeper issues. You just set yourself up for the same puzzle at the next geometric level — apokrisis

I'm not saying that 1D lines are the fundamental object and all other objects are constructed from them. As you say, we run into the same issue when constructing 2D surfaces from 1D curves. I am proposing that instead of constructing the whole from the parts, that we construct the parts from the whole. We start with the highest dimensional continuum of interest. Think about how we draw a graph on paper. We don't make use of pointillism. We draw a square on our paper and then cut up that square by drawing some lines, for example the line x=0 and y=0. Only when these lines are drawn does the point (0,0) emerge. The more lines we draw, the more points emerge. Might the same apply to objects in the abstract world? Might continua be fundamental instead of points?

I mean it doesn't even make sense to talk about 0D points except in the context, or in contrast, with the presence of the 1D line, right? — apokrisis

But conversely, we can talk about a 1D line in the absence of points. In the example above, when I draw the first line and assign to it the function x=0 I haven't drawn any points yet. All I've done is draw a line and described how that line will interact with other lines IF they are added to my drawing. And I can keep adding more lines to my drawing to no end, adding more and more points, but never will I have infinite points. Or in other words, no matter how many times I cut up a piece of paper, never will it vanish to nothingness. Never could a continuum be decomposed into points. For this reason, I have to disagree with you when you say that continua as foundational objects are offensive to the ontic intuition.

This would see the discrete and the continuous as being each others limiting case. — apokrisis

The thing is that we can't go the limit. We can't complete the computation of the infinite series 0+0+0+0+... but as far as I can tell it looks like it should add to 0. I find it hard to believe that in some cases it adds to 1 and in other cases it adds to 2. But this is what we're doing when we assemble 0-length points to create lines of length 1 and 2.

Instead, to me it makes more sense to start with a continuum. Start with a string of length 10. Cut it in half, then you get strings of length 5. Cut one string in half, then you get two strings of length 2.5. Keep going and as you go strings of potentially infinite different lengths emerge. There's substance there from the start. You don't have to go the limit to have useful objects when starting with a continuum. With this parts-from-whole construction, objects are finite and processes are potentially infinite...and there are no paradoxes. This is in direct contrast to set theory where the whole is constructed from the parts and the objects of study (sets) are actually infinite.

Why can't we just say that pi is not a number? Instead, it is an algorithm (e.g. pick your favorite infinite series for pi) used to generate a number. This algorithm is potentially infinite in that we can never complete it, but we can certainly interrupt it to generate a rational number. If you interrupt it, maybe you'll get 3.14. Actual infinity only comes into play if you claim that the algorithm can be completed, in which case it would generate a real number - a number with actually infinite digits. This is what I would like to challenge. — keystone

I don't understand why you want to challenge this. I use approximations to pi all the time. When I want a quick and dirty approximation of the area of a circle inscribed in a square with sides x, I use 3/4 * x^2. I can round pi off anywhere I like depending on the precision I need. To say that irrational numbers are not really numbers doesn't make any sense to me. Of course they are.

Perhaps I should have written that I believe it is impossible to imagine assembling points to form a continuum. — keystone

I really don't get this. I have no problem imagining continuity arising from discreteness. I learned, saw it, got it, in 6th grade algebra. As @apokrisis wrote in a later post:

We need a more subtle metaphysics. We need an intuition that itself sees parts and wholes, the discrete and the continuous, as the two emergent parts of the one common rational operation. — apokrisis

Holding two apparently contradictory ideas in your mind at the same time is a required skill, e.g. waves and particles. It's no big deal. I learned that, saw that, got that in 12th grade physics.

What advantage is there in seeing things your way. Expecting abstract concepts such as mathematical entities to have some sort of ontological reality doesn't make sense. Mathematicians love math for math's sake. Engineers such as me just want something that works - no ontological interpretation necessary. I assume the same is true for most scientists. How does your way work better than the way it is handled normally?

Like many who are philosophically inclined, I am happy to accept actual infinities as a useful mathematical simplification – an epistemic trick – but not something that makes proper ontological sense. — apokrisis

It makes ontological sense to me. I do agree that is a useful, abstract simplification. Really, all math is. All reality is.

So the idea of 0D points – some kind of absolute notion of discreteness – is offensive to the ontic intuition. But the same should apply to its dichotomous "other", the idea of an absolute continuity as the alternative.

We need a more subtle metaphysics. We need an intuition that itself sees parts and wholes, the discrete and the continuous, as the two emergent parts of the one common rational operation. — apokrisis

I can imagine these apparently contradictory ways of seeing things could be difficult to grasp, but it's something you need to do if you want to use math. As I said, from the beginning, I could see that resolving this dichotomy is inherent in understanding all of math, at least the more practical math that engineers and scientists use.

What does this mean for number lines? It says that while we must think of the 1D whole being constructed of 0D points, that claim must be logically yoked to its "other" of each 0D point existing to the degree the 1D continuity of the line has in fact been constrained. — apokrisis

Agreed.

I mean it doesn't even make sense to talk about 0D points except in the context, or in contrast, with the presence of the 1D line, right? — apokrisis

Right.

I am proposing that instead of constructing the whole from the parts, that we construct the parts from the whole. — keystone

Yep. So construction gets replaced by constraint. And then my point is you go the next step of seeing construction and constraint as the two halves of the one system. Which is where you arrive at a triadic metaphysics – the one where a vague potential gets organised by the reciprocal deal of construction~constraint.

We start with the highest dimensional continuum of interest. — keystone

Which would be the "infinite dimensional" continuum ... unless you can find some larger argument that tells you what actually regulates the emergence of algebraic structure.

So when algebra is placed under the constraint of having to preserve normed division, you get the reals, complex, complex, quartonions, octonions and even the 16D sedonions. But you also get a petering out of the mathematical properties you are trying to preserve.

To me, that is a good structuralist argument. You might think any number of dimensions might still contain algebra. Or you might think that only one dimension contains "real algebra". Well actual algebra acts as a constraint that gives you something intrinsically more complex when it is run over the whole space of what seems possible.

And quantum theory even suggests complex numbers as the true centre for algebraic structure – as it makes commutativity count out in the real world of particles and symmetry breaking.

So I am saying I wouldn't deal with the metaphysics of the number line in isolation. It is illustrative of the far bigger conversation we need to have about how holism in mathematical conception plays out. The same principles have to cover mathematical structure in general – as category theory argues, having absorbed the metaphysics of Hegel and Peirce.

Might the same apply to objects in the abstract world? Might continua be fundamental instead of points? — keystone

This is a rather basic level of discussion. Again, how could it even be a continua unless it could be cut? How could it even be a 0D point except as the positive absence of any dimensioned extension?

I draw attention to the fact that you want to make one thing "the fundamental". This is monism. This is reductionism. This is not holism.

Holism says everything is a system of relations. And so the first number you need to get to is two. Some pair of "fundamental things" in relation. This in turn only makes sense when you get to three "fundamental things" in relation.

You have to get to the point where the relating of things is itself dualised in some limit case fashion. You have to arrive at a metaphysical dichotomy or reciprocal relation where one of the things is your local scale of being – your world-constructing collection of individual parts – and the other is then your global scale of being, or the constraints, habits, laws, emergent macro-properties, downward causality, etc, that is the generalised holism of the system in question.

So if you want to apply the strength metaphysics to questions about mathematical structure, you have to count to three in terms of "fundamental things". But fortunately three is then enough. Its a theorem in network theory that all networks of relations reduce to "threeness". :grin:

Or in other words, no matter how many times I cut up a piece of paper, never will it vanish to nothingness. — keystone

But each piece also gets more pointlike. So eventually the matter becomes obscured by arriving at that even more fundamental thing of being a vagueness.

The cut has to be sandwiched between the two ends of two lines. Each end of the line is a point. At what point does the point marking the cut – that is, the absence of a point at that point – get marked off from the other two points marking the starts of a pair of now separated continua?

You will be familiar with these kinds of arguments. And they make no real sense because they talk about dimensionless points and dimensioned lines without any clear definition of the relation between the two. There is no operation connecting them.

But if you have an argument based on a reciprocal relation, then each only exists to the extent it is a limitation its other. And that is how you recover the intuitionist ontology of Brouwer.

An infinite amount of cutting will result in an infinite number of number line pieces. And an infinite amount of gluing will put the number line back together.

But that then means you have to be able to both cut and glue. The two actions go hand in hand. If either action runs out of steam, so does its "other".

So it is easy to picture just forever cutting a line. Or instead, just forever gluing points. Yet both are equally one-eyed perspectives in a mathematical reality where this has to be in fact a reversible operation. The two sides of the equation need to be resepcted by our ontic interpretation.

And what do we get when we roll back the reciprocal operation of cutting~gluing back to its primal origin where it is first detectable as a structure-forming relation?

Points and lines are what we see by the end in a rather black and white fashion. But what were they as some kind of distinction of this type first started to swim into view?

The thing is that we can't go the limit. — keystone

But the fact that we can approach the limit – both limits – with arbitrary closeness is how we know they are there. The limit is precisely that which isn't reachable in the end. But it certainly defines the direction we need to keep going from the start.

And this only makes sense if we are starting from the symmetry breaking of a dichotomy. The initial split is simply a reaction against the logical "other".

So each limit is defined as heading in the opposite direction of what is being left behind. Continuity is the impulse to put as much distance from discreteness as possible. Discreteness is likewise the intent of becoming as discontinuous as can be imagined.

So at the start of things – the foundational conditions which is a logical vagueness – the need is for a direction that leaves something behind. So there is the need for two things in fact. And both of them are trying to leave each other behind. If there are more than two things trying to leave each other thing behind then nothing really gets left behind in a maximal or extremitising way. Again, that is the definition of a dichotomy – mutually exclusive and jointly exhaustive.

Thus logic can define the start of a distinction in terms of a reciprocal desire to simply separate. From there, both sides will go as far as they can go towards matching limits. But because this dividing is only real for as long as distance is being created between the two, then neither can actually become separated "in the limit" as they remain yoked together by their need for this opposition.

The point and the line, or the infinitesimal and the infinite, are actualised to the degree they are actively divided.

With this parts-from-whole construction, objects are finite and processes are potentially infinite...and there are no paradoxes. — keystone

Again, this suffers all the usual problems of an object-oriented ontology. Reality is better understood in terms of relations – processes and structures.

objects are finite and processes are potentially infinite — keystone

This may be true, but I don't think everybody qualified to have an opinion agrees with you. There are physicists who believe the universe is infinite. That doesn't really make sense to me, but a lot of things that don't make sense to me turn out to be true, so I'll remain agnostic.

objects are finite and processes are potentially infinite — keystone

Most interesting — Ms. Marple

$\infty$ isn't and object like for example an elephant or the number 10¹⁰⁰ or the word "elephant", it's simply a shorthand for the procedure 1. n = 0; 2. print n; 3. n = n + 1; 4. go to 2]. :chin:

That can't be true. Calculus is all about infinity — T Clark

I was thinking the same thing.

That can't be true. Calculus is all about infinity — T Clark

I was thinking the same thing. — god must be atheist

Once again, calculus is about LIMITS, as my mathematical genealogical ancestor, Karl Weierstrass would have explained.

That damn lemniscate and the problems it produces . . . :roll:

I have a feeling that some ideas like $\infty$ and nothing cause brain damage - Cantor lost his mind (theia mania) and spent his later years in a lunatic asylum for instance. These concepts & paradoxes of which there are many seem to have a deletorious effect on the brain/mind - constantly mulling over them may lead to a nervous breakdown. Such ideas are more than our brains can handle at present. And yet ... there have been no reports of an epidemic of mental problems among mathematicians. Why I wonder.

There are physicists who believe the universe is infinite — T Clark

With circular reasoning. Perhaps a label for endless but not quite infinite in a physical sense ?

Never could a continuum be decomposed into points — keystone

For physics, isn't that the driving force behind quanta, to put a stopper into space leaking out ?

Once again, calculus is about LIMITS, as my mathematical genealogical ancestor, Karl Weierstrass would have explained. — jgill

Okay. Try this:

An object is at rest. It is not moving.

Now the object is moving at a velocity V.

How many different velocities did the object move at, to get from zero velocity to V velocity?

If your answer is not "infinite" then you don't deserve the name "mathematician". Because calculus presupposes that there are infinite velocities there.

Get off your high horse.

Once again, calculus is about LIMITS, — jgill

True, but in many a calculus problem and theorem the limit IS infinity.

I don't understand why you want to challenge this. I use approximations to pi all the time. When I want a quick and dirty approximation of the area of a circle inscribed in a square with sides x, I use 3/4 * x^2. I can round pi off anywhere I like depending on the precision I need. To say that irrational numbers are not really numbers doesn't make any sense to me. Of course they are. — T Clark

You say there exists a number called pi with infinite digits and you use a truncated approximation of it when you calculate the approximate area of a circle.

I say that what exists is a (finitely defined) algorithm called pi that doesn't halt but you can prematurely terminate it to produce a rational number to calculate the approximate area of a circle.

The difference is that you are asserting the existence of an infinite object, something beyond our comprehension. My approach seems more in line with what us engineers actually do, so why bother asserting the existence of something impossible to imagine if you don't even need to?

I really don't get this. I have no problem imagining continuity arising from discreteness. I learned, saw it, got it, in 6th grade algebra. — T Clark

Do you believe that 0+0+0+0+... can equal anything other than 0? If not, then how can you claim that 0-length points could be combined to form a line having length?

Holding two apparently contradictory ideas in your mind at the same time is a required skill, e.g. waves and particles. It's no big deal. I learned that, saw that, got that in 12th grade physics. — T Clark

Sounds like double-think from 1984. There are no contradictions in wave-particle duality.

What advantage is there in seeing things your way. Expecting abstract concepts such as mathematical entities to have some sort of ontological reality doesn't make sense. Mathematicians love math for math's sake. Engineers such as me just want something that works - no ontological interpretation necessary. I assume the same is true for most scientists. How does your way work better than the way it is handled normally? — T Clark

With my view many paradoxes (Zeno, Dartboard, Liar's, etc) are easily resolved. While this should be enough, it also aligns far better with what us finite beings actually do. Also, while I haven't gotten into it here, it makes quantum physics less weird. Ultimately, I'm talking about the philosophy of mathematics, not the application of it. The day to day mathematics of engineers don't change with this new foundation.

It makes ontological sense to me. I do agree that is a useful, abstract simplification. Really, all math is. All reality is. — T Clark

Yes, all reality is void of actual infinities. So why do we need to believe that reality is just an approximation of some ideal infinity-laden object that we can't comprehend or observe? Why can't we stop at reality?

This may be true, but I don't think everybody qualified to have an opinion agrees with you. There are physicists who believe the universe is infinite. That doesn't really make sense to me, but a lot of things that don't make sense to me turn out to be true, so I'll remain agnostic. — T Clark

They think it's possible only because modern math welcomes actual infinity. If mathematicians rejected actual infinity then I'm sure physicists would be less inclined to accept it.

Can you explain this to me from a computer programming perspective? In your comparison, is the data the output of the function? A function can return a function, but it can also return another object type, like a string. In the latter case, there is a type distinction between between the function and its output, but I don't see how this is unnecessarily rigid. I suspect I'm missing your point. — keystone

I'm basically warning against logicism, the ideology that there is a single correct logical definition of a mathematical object. Thinking in this way leads to unnecessary rejection of infinite mathematical objects, for such objects aren't necessarily infinite in a different basis of description. e.g the length of a diagonal line doesn't have infinite decimals relative to a basis aligned with the diagonal.

Also, the algorithm for approximating sqrt(2) to any desired level of accuracy can itself be used to denote sqrt(2) without being executed.