Just out of curiosity though, how do you develop a theory of the real numbers without infinite sets? Even the constructivists allow infinite sets, just not noncomputable ones......The problem with finitism is that you can't get a decent theory of the real numbers off the ground. — fishfry
If I see a coherent one presented I'll engage with it. In the past I've engaged extensively with constructivists on this site and learned a lot about the contemporary incarnations of that viewpoint. I've also studied the hyperreals of nonstandard analysis. So in fact I'm very open to alternative versions of math, but I don't see that you've presented one. — fishfry
We measure the car at 60mph and maybe that's accurate to within a small margin of error. But at no point during its 60 mph run is its speed zero. — tim wood
It seems to me that with any line I look at I'm looking at an infinite number of points. Not potential points, but actual points - that is, to the degree that one, or any, point is actual. — tim wood
There is an issue of truth here. There is something there causing the form, and the concept of "field" attempts to account for whatever it is. If the concepts employed are inadequate, then it's not true to say that this is what is there. — Metaphysician Undercover
If the process is terminated then it is untrue to say that it is potentially infinite. — Metaphysician Undercover
And if we know that in every instance when such a process is useful, it is actually terminated, then we also know that it is false to say that a potentially infinite process is useful, because it is only by terminating that process, thereby making it other than potentially infinite, that it is made useful. — Metaphysician Undercover
In line with Aristotle's solution to Zeno's Paradox, my view has continua (not points) being fundamental. I think the mathematics of calculus would be almost entirely unaffected in moving to a continuum-based view, — Ryan O'Connor
Why isn't Aristotle's solution just circular because it makes the results of a mathematical construction prior to the construction itself? — magritte
If I give up points as bounds, then how would I have anything but an endless line? — magritte
I think this is the wrong question to ask. We know what we want (a consistent foundation for calculus) and we think we know how to get it (with real numbers and infinite sets). But real number might not be the answer.
I think a much better question to ask is 'can we build a consistent foundation for calculus without real numbers and infinite sets?' I believe the answer is yes. Now, this doesn't imply that the alternative requires scrapping everything about real numbers. For example, I believe the alternative would have still have the area of a unit circle being pi, it would just mean something different. — Ryan O'Connor
Like the real numbers, the fundamental objects of the hyperreals are points/numbers. A continuum is constructed by assembling infinite points, each with a corresponding number. And so the hyperreals gets no closer to answering the question 'how can a collection of points be assembled to form a line?' I haven't investigated the constructivists' methods to produce an informed comment but from what I've seen it looks like more of the same. Points/numbers are treated as fundamental.
I don't have a formal theory but I do have an intuition on how the alternative would look. In line with Aristotle's solution to Zeno's Paradox, my view has continua (not points) being fundamental. — Ryan O'Connor
I think the mathematics of calculus would be almost entirely unaffected in moving to a continuum-based view, but before we could even talk about it, you'd have to first be open to (at least temporarily) shedding your point-based biases and look at graphs in a new light. — Ryan O'Connor
I would love to hear your criticisms on my earlier post to you where I drew the polynomial in an unorthodox way. — Ryan O'Connor
That leaves the question, when there is no interval of time, is it meaningful to speak of time or anything that requires time. And this moves us towards the idea of a limit. Not as the thing itself, but as the thing approached.But this doesn't imply that I believe that there is some interval for which the car's average velocity is 0mph. — Ryan O'Connor
I've looked into constructivism a bit, which is enjoying a modern resurgence due to the influence of computer science and computerized mathematical proof assistants. But you would reject even that. — fishfry
But why am I supposed to be burdened for finding a mathematical viewpoint other than the standard one accepted by almost all the world's mathematicians except for those pesky constructivists and rare finitists and ultrafinitists? Why is this my problem, or math's problem? — fishfry
I suppose I owe you that. I'm going to find it depressing and frustrating but I'll take a run at it sometime. — fishfry
A program written to spit out the natural numbers one at a time is potentially infinite, regardless of whether it's been executed or interrupted. — Ryan O'Connor
If you have ever seen π as the solution to a problem (instead of, say, 3.1415) then the process hasn't been terminated, it hasn't even been initiated. It's incorrect to say that potentially infinite processes are only useful when prematurely terminated. — Ryan O'Connor
That leaves the question, when there is no interval of time, is it meaningful to speak of time or anything that requires time. — tim wood
If it makes sense to speak of a time as a "time" when there is no motion, then that same notion applied to a car implies that it's spending most of its time, even while in motion, not in motion. — tim wood
We already discussed the difference between the rule ("program" in this case) which sets out, or dictates the process, and the process itself. If the process is interrupted, it ends, and is therefore not infinite. The rule ("program") is never infinite, nor is it potentially infinite, it's a finite, written statement of instruction, like "pi", and "sqrt (2)" are finite statements, even though they may be apprehended as implying a potentially infinite process. — Metaphysician Undercover
I don't see how you can say this. Pi says that there is a relationship between a circle's circumference and diameter. This information is totally useless if you do not proceed with a truncated version of the seemingly infinite process, such as 3.14. "The solution to the problem is pi" doesn't do anything practical, for anyone, if you cannot put a number to pi. — Metaphysician Undercover
even pi can only be understood as part of a finite number — Gregory
I certainly like the idea of constructivism in that it is necessary to construct a mathematical object to prove that it exists — Ryan O'Connor
-The program spits out numbers as it is being executed, so it doesn't need to be terminated to get something useful from it. — Ryan O'Connor
-We can discuss the execution of the program without ever running it (e.g. we can say 'if I executed the program, it would be potentially infinite) — Ryan O'Connor
n the end, I think you're splitting hairs here. What's your point? — Ryan O'Connor
π is often written as the solution to a problem - for one it's what they say is the volume of Gabriel's Horn. — Ryan O'Connor
Also, who said math had to be practical? — Ryan O'Connor
In line with Aristotle's solution to Zeno's Paradox, my view has continua (not points) being fundamental. — Ryan O'Connor
I have not had the time, energy, or patience to jump into the substance of this discussion so far, but I can offer a couple of reading suggestions based on these two comments.Sounds like a Peirceian viewpoint, about which I don't know much. — fishfry
I think a much better question to ask — Ryan O'Connor
The second is my own recent paper, — aletheist
Your ontology is weak. You say a table is one, yet it has 4 legs and a top piece. What number of parts do these have? This process is infinite and it takes a delicate balance to understand all it's intricacies. Motion passes through infinity and the finite, but you want to reduce the question to Aristotle's lame argument,: namely that parts are only potentially there. Bringing in QM isn't going to help your case mr. idealist. The world is real. "Ignore the world and the world will come to you" — Gregory
But what good is that? (in reference to talking about a potentially infinite process) — Metaphysician Undercover
This is probably the crux. "Math does not have to be practical". — Metaphysician Undercover
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