keystone

I disagree. I think that potential infinity comes naturally to us, whereas we reluctantly hold on to actual infinity. We do so because we are not used...

Must an ordered field necessarily be a field of numbers? Could it instead be a field of equations? For example, I would imagine that adding a 1 to eac...

ELI5. You believe that math itself has some fundamental limits, perhaps a frequency, a duration, or a length. You may be right but I think you're wron...

I understand how we got here, it's just that to me this screams that what got us here is a mistake. I have a hard time accepting real numbers, I have ...

Undefinable real numbers have no place in my view. Focusing solely on definable irrationals, I believe an irrational is irrational, no matter how effi...

Please allow me to indulge in Hilbert's Hotel a little more just so that we can stick with actually infinite objects instead of potentially infinite p...

I believe that, contrary to the mainstream interpretation, 19th century analysis is the mathematics of continua (not the mathematics of points). I do ...

I think it's quite easy to imagine a closed finite universe, for example a sphere of finite radius. It's a lot harder to fit in one's mind a sphere of...

I would say that those paradoxes don't necessarily impugn set theory itself. Just as actual infinities (rather infinitesimals) were banished from (mai...

One would be crazy to say that classical math is wrong. I'm in no position to say this with certainty, but I believe that our finite descriptions of a...

My impression is that we do not prove sqrt(2) is a number, but instead we assume it is a number by means of the completeness axiom.

In terms of using the truths of physics as a guide in the hunt for truth in mathematics, I'm totally with you. However, I feel like you're mentioning ...

The above is correct.

If the infinitistic systemization for mathematics are more powerful, beautiful, and simple then there's not much appeal to a non-infinitistic systemiz...

In Hilbert's full Hotel, move guest 1 to room 2, guest 2 to room 3...yada yada yada...it can accept an additional guest. Nobody explained the paradox ...

I'm not sure what you're referring to.

I will have to trust you on your claims related to ZFC since I don't have a good understanding of it. What was the 19th century analysis resolution to...

I know one can easily imagine imagining the hotel (i.e. that above each floor there is another floor) but can you imagine the actual endless hotel as ...

Returning to my magic eye analogy, the image pops up only because the background does not. Can the image and background be the 'measure of the other' ...

It's not about me. It's about computers in general. I can imagine a computer picturing a 4D hypershere as easily as a sphere. And if you don't accept ...

I would argue that the 'real number line' should instead be called the 'real line' since it's composed of more than just numbers. Consider the proof t...

I agree about the importance of the real world, and perhaps investigating the sub Planck scale is important for the deepest insights. I just don't thi...

I should have explained explicitly what I meant when I wrote "(Cantor)" as you interpreted my intention backwards.

Wikipedia: The set of natural numbers (whose existence is postulated by the axiom of infinity) is infinite. It is the only set that is directly requir...

Why can't we talk naively about points combining to form a line? It seems a little disturbing that to discuss what is seemingly a very simple concept ...

I agree. IF there is any merit to my view, then the hard work hasn't even begun.

Most notably Hilbert's paradox of the Grand Hotel, but also the following: Gabriel's horn Galileo's paradox Ross–Littlewood paradox Thomson's lamp Zen...

Is it possible for a continuum to exist and be defined mathematically without relying on numbers? I'm referring to a curve (1D continuum), surface (2D...

No, I don't believe the continuum is everything. I think that the computer/mind lies outside the continuum. For example, when you imagine a sphere you...

I agree that in terms of the orthodox point-based interpretation of calculus that there would be infinite points along a line. However, a continuum-ba...

I think it boils down to the real line being composed of points and each point having a unique number. Instead of focusing on pi, let's return to the ...

Let's imagine a line where cuts have been made to mark all rational points (I don't believe this is possible, but let's go with it for now). I believe...

@"apokrisis" I'm just rereading through your older messages and I have a few additional comments: I see neither the point as infinitesimal nor the lin...

This is true, and I sincerely appreciate everyone's responses.

What I'm saying is that pi isn't the string of digits that begin with 3.14. Pi is the algorithm(s) such as the infinite series beginning with 4/1 - 4/...

I really like Norman Wildberger. I think his issues with the foundations of mathematics are valid. However, I do not like his resolution to the issues...

I see. I'm not really heading down the logicism route at the moment. Agreed. That is why I what to give existence to the algorithm and not the complet...

In a quantum reality we can only talk about it's velocity when measurements were made. Since we can only ever make a finite number of measurements in ...

Somewhat related, my understanding is that the planck length applies to measurement, not space itself. My understanding is that the continuity of spac...

I agree. However, I would go one step further and say that infinity-laden objects such as real numbers are also not objects like the number 10100. Ins...

The need to complicate things by forming your triadic metaphysics is unclear to me. You say that the idea of an absolute continuity as the alternative...

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 ...

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 c...

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 ...

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 ...

Maybe not as challenging as you think. I think the application of real numbers would remain largely unchanged, it just requires a reinterpretation. If...

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....

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 imag...

Thanks @"Andrew M"! You’ve cleared up much of my confusion. I understand that a lot of the mystery of the DCQE experiment can be addressed by fully ap...

No wait…the first entangled photon’s behaviour is random but once measured, the other photons behaviour is determined? So in the DCQE are you saying t...