OK, so we agree that if so-called "mathematical objects" are things which can be measured, Euclidian geometry creates distances which cannot be measured by that system. That agreement is a good starting point. — Metaphysician Undercover
OK, so we agree that if so-called "mathematical objects" are things which can be measured, — Metaphysician Undercover
Euclidian geometry creates distances which cannot be measured by that system. — Metaphysician Undercover
As a philosopher, doesn't the question, or wonderment, occur to you, of why we would create a geometrical system which does such a thing? — Metaphysician Undercover
That geometrical system is causing us problems, inability to measure things, by creating distances which it cannot measure. — Metaphysician Undercover
Maybe we can take this as another point of agreement. A "mathematical object" is nothing other than what you called a "funny gadget". Let's simplify this and call it a "mental tool". Do you agree that tools are not judged for truth or falsity, they are judged as "good" in relation to many different things like usefulness and efficiency, and they are judged as "bad" in relation to many different things, including the problems which they create. — Metaphysician Undercover
So a "good" tool might be very useful and efficient, but it might still be "bad" according to other concerns, accidental issues, or side effects. — Metaphysician Undercover
Bad is not necessarily the opposite of good, because these two may be determined according to different criteria. — Metaphysician Undercover
Let's look at the Euclidian geometry now. In relation to the fact that this system produces distances which cannot be measured within the system — Metaphysician Undercover
, can we say that it is bad, despite the fact that it is good in many ways? — Metaphysician Undercover
How should we proceed to rid ourselves of this badness? — Metaphysician Undercover
Should we produce another system, designed to measure these distances, which would necessarily be incompatible with the first system? — Metaphysician Undercover
Having two incompatible systems is another form of badness. Why not just redesign the first system to get rid of that initial badness, instead of creating another form of badness, and layering it on top of the initial badness, in an attempt to compensate for that badness? Two bads do not produce a good. — Metaphysician Undercover
Come on, get real fishfry. Check Wikipedia on set theory, the first sentence states that it deals with collections of "objects". — Metaphysician Undercover
Then it goes on and on discussing how set theory deals with objects. Clearly set theory assumes the existence of objects, if it deals with collections of objects. — Metaphysician Undercover
This is why it is so frustrating having a conversation with you. You are inclined to deny the obvious, common knowledge, because that is what is required to support your position. — Metaphysician Undercover
In the other thread, you consistently denied the difference between "equality" and "identity", day after day, week after week, despite me repeatedly explaining the difference to you. — Metaphysician Undercover
You have not explained how acceptance of a mathematical tool, through convention, converts it from a funny gadget, to an object. — Metaphysician Undercover
If you cannot demonstrate this conversion, then either the tool is always an object, or never an object. — Metaphysician Undercover
Then an extremely bad tool is just as much an object as an extremely good tool, and acceptance through convention is irrelevant to the question of whether the mental tool is an object. — Metaphysician Undercover
Until you recognize that an "element", or "member" of a set is an "object", you are simply in denial of the truth, denying fundamental brute facts because they are contrary to the position you are trying to justify. — Metaphysician Undercover
The case I made is very clear, so let me restate it concisely. You appear to agree with me that mathematical tools are not objects, they are "mind" gadgets, yet you defend set theory which treats them as objects. — Metaphysician Undercover
This is nonsense. I can very easily say "the highest number". Just because I say it doesn't mean that what I've said "completely characterizes" it. We can say all sorts of things, including contradiction. Saying something doesn't completely characterize it. — Metaphysician Undercover
if you switch to a different number system, one which is incompatible with the first from which the irrational number is derived, like switching from rational numbers to real numbers, this does not qualify as a resolution, if the two systems remain incompatible. — Metaphysician Undercover
For instance, if there is infinite rational numbers between any two rational numbers, and we take another number system which uses infinitesimals or some such thing to limit that infinity, we cannot claim to have resolved the problem. The problem remains as the inconsistency between "infinite" in the rational system, and "infinitesimal" in the proposed system. — Metaphysician Undercover
This has no relevant significance. To say "the square root of two", or "the ratio of the circumference of a circle to its diameter" is to give a 'finite description". We've already had the "finite description" for thousands of years. And, this finite description determines that the decimal digits will follow a specific order, just like your example of 1/3 determines .333.... The issue is that there is no number which corresponds to the finite description, as is implied by the infinite procedure required to determine that number. — Metaphysician Undercover
So my analogy of "the highest number" is very relevant indeed. Highest number is a "finite description". And, the specific order by which the digits will be "computed" is predetermined. However, there is no number which matches that description, "highest number", just like there is no number which matches the description of "the square root of two", or "the ratio of the circumference of a circle to its diameter", or even "one third". — Metaphysician Undercover
This demonstrates that there is a problem we have with dividing magnitudes, which has not yet been resolved. — Metaphysician Undercover
Let me return your attention to this remark. If you agree with me, that the representations are "imperfect" from the start, then why not agree that we ought to revisit those representations. Constructing layer after layer of complex systems, with the goal of covering over those imperfections, doing something bad to cover up an existing bad, is not a solution. — Metaphysician Undercover
By "models' factorizations" I mean finding the right definitions that allow you to describe some complex (containing a lot of information) models in a simple way, or that allow you to prove something that was too complex to prove without these definitions. . . . . That's mainly what mathematicians are doing today — Mephist
You keep clinging to your mistaken belief, thinking that the rational numbers are good and the irrationals bad. This is a personal psychological condition that can be remedied with mathematical knowledge. If you so desire. — fishfry
You keep repeating this without engaging with the fact that math says you're wrong. — fishfry
I am not responsible for what people type into Wikipedia. Some math articles are very good, some are highly misleading. — fishfry
I'm speaking sophisticated math to you and you just want to cling to what they told you in high school. That's your choice. — fishfry
I did. The passage of time. — fishfry
It's something you made up. — fishfry
You are psychologically stuck to your wrong ideas and you're incapable of engaging substantively with any point that anyone makes. — fishfry
ou are really good at writing stuff that sort of sounds intelligent, but doesn't hold up to scrutiny. That's why I was initially interested in your posts and why I took the trouble the read them. Now I've scrutinized them. You're ignorant about math and wrong on the philosphy. — fishfry
That's right! Rational numbers are just as fictional or just as real as irrational numbers. — fishfry
he bad thing is some misinformation you got stuck with in high school or earlier. You have to let go of things you think you learned that don't happen to be true. — fishfry
learned a lot talking with you. Mostly I learned that I know a lot more about the philosophy of math than at least one person on this site who claims to know a lot about the philosophy of math. It's taken me years to get to this point. Thank you. — fishfry
Fishfry! Get with the program, wake up and smell the coffee! — Metaphysician Undercover
it's pointless to speak sophisticated math at me. — Metaphysician Undercover
My favorite part of the day is my morning coffee. — fishfry
Ignorance as a debating point. "Your argument stands refuted because I'm incapable of understanding it." I can't top that. — fishfry
The problem is that you do not address the substance of the argument. You go off on some tangent using mathematical jargon, without addressing the issue. — Metaphysician Undercover
You can literally build a square root of 2 out of the rational numbers. — fishfry
The construction procedure you described is never ending, just like the never ending digits. — Metaphysician Undercover
* Now the set {a+b2–√:a,b∈Q}{a+b2:a,b∈Q} happens to have a mathematical structure identical to that of the rationals; and in addition, it contains a square root of 2. — fishfry
Fact: If you believe in the rationals, you must believe in the rationals augmented by the square root of 2. — fishfry
If you are going to make a mathematical claim you need to be able to accept a mathematical disagreement. — fishfry
Why not move on, and inquiry what this principle tells us about numbers and spatial relations, instead of trying to disprove it. — Metaphysician Undercover
It appears like we need to go back over the law of identity, and the difference between identical and equal. Remember, I don't accept set theory on the basis that it violates the law of identity, so why give me a proof based in a set? — Metaphysician Undercover
I don't accept set theory on the basis that it violates the law of identity — Metaphysician Undercover
This means that taking a square root is not a valid operation — Metaphysician Undercover
I am curious to know: do you have an answer to this question? — Mephist
You want to make a mathematical claim (sqrt 2 doesn't exist) but you won't accept a mathematical response. Makes for pointless conversation. — fishfry
I have already explained to you at length that set theory is based on the law of identity; and that the mathematical equals sign expresses identity between two expressions. — fishfry
Define "valid operation." You should have been around to make your current argument about 1700BC when the Sumerians were calculating the square root of two (and its reciprocal) on cuneiform tablets. They would have appreciated your perspective. — jgill
Your solution involves a violation of the fundamental laws of logic, the law of identity (as explained on the other thread), therefore I reject it. — Metaphysician Undercover
I think there are two issues becoming evident. One is that we do not know how to properly represent space. The irrational nature of the "square", and the "circle", as well as the incompatibility between the "point" and the "line" indicate deficiencies in our spatial representations. — Metaphysician Undercover
The other is that we do not know how to properly divide something. There is no satisfactory, overall "law of division", which can be consistently, and successfully used to divide a magnitude. We tend to look at division as the inversion of multiplication, "how many times" the divisor goes into the dividend. Because there is often a remainder, division really cannot be done in this way. The "square root of two" is a more complex example of this simple problem of division, the issue of the remainder. — Metaphysician Undercover
To state the problem succinctly, set theory allows that two distinct things have the same identity, in the same way that we might say two distinct things are equal. The faulty premise is that things with the same value "2" for example, are the very same thing. In other words set theory premises that, "2" refers to an object, rather than a value assigned to an object. It is a category mistake to treat what "2" refers to, as a particular object, rather than as a universal principle. — Metaphysician Undercover
Well, the "issue" of the irrationality of the diagonal of the square is the one that ancient greeks recognized: you cannot find any unit length that enters both in the side and in the diagonal of the square an integer number of times (no matter how little you take your unit length).
So there cannot exist any fundamental minimal length of physical space — Mephist
Try to find M and N such that the sticks arrive at the same point. Since M/N is irrational, you can't do it, — Mephist
So there cannot exist any fundamental minimal length of physical space (kind of a microscopic indivisible stick) that can be oriented in any direction. If there is such a thing, every physical object at the microscopic scale should be made of tetrahedrons, or something similar. So circles and squares are really only approximations of the real "physical" shapes. Is your idea something of this kind? If not, in what other way can you make all the lengths be rational numbers? — Mephist
You mean that there is no defined physical procedure to divide a generic geometrical segment by another? — Mephist
If you take two generic segments of whatever length, you can always build a third segment that is proportional to their ratio (whatever it is, even irrational). — Mephist
Sorry for the intrusion, but I am curious of this issue (only one premise: I didn't study philosophy :yikes:, so, for example, I don't really understand why this "law of identity" is so important...).
However, that's my question: how do you refer to an object instead of to it's value? I mean: if every symbol refers to a different object, even if the symbol is the same as the one that you used before, you can never refer to the same object twice, can you? — Mephist
By no one. The aether was always supposed to have been something in space, that gave space properties that empty space, supposedly, did not have. You're losing it, MU, and I'd be sympathetic, but your dogmatism won't allow it. Instead you become a joke, laughable, and that's a shame, because if memory serves, that did not use to be the case.Once it was believed that space is an aether, — Metaphysician Undercover
And,What I am talking about specifically, is dividing numbers. Divide ten by three, and you have a remainder of one. It is the remainder which is a problem. When we multiply numbers we never get remainders, yet we tend to treat division as the inversion of multiplication. It's actually quite different from multiplication because multiplication starts from premises of fundamental base units, whereas division presupposes no such base units.. So I think we need to pay close attention to this fact, that constructing a magnitude through multiplication is really a completely different process from destroying a magnitude through division. — Metaphysician Undercover
Once it was believed that space is an aether, so the fundamental shapes were waves. A wave is active, so it requires the passing of time, for its activity. So let's assume "space" is an active medium. — Metaphysician Undercover
You're losing it, MU — tim wood
A wave is active, so it requires the passing of time, for its activity. So let's assume "space" is an active medium. Now suppose we try to make something static, like a circle or a square, within this medium which is active. The shape won't actually be the way it is supposed to be, because the medium is actively changing from one moment to the next. So if we want to make our shape, (circle or square), maintain its proper shape while it exists in an active medium, we need to determine the activity of the medium, so that we can adjust the shape accordingly. Understanding this activity would establish a true relationship between space and time, because defining this activity of space would provide us with a true measure of time. — Metaphysician Undercover
What I am talking about specifically, is dividing numbers. Divide ten by three, and you have a remainder of one. It is the remainder which is a problem. When we multiply numbers we never get remainders, yet we tend to treat division as the inversion of multiplication. It's actually quite different from multiplication because multiplication starts from premises of fundamental base units, whereas division presupposes no such base units.. So I think we need to pay close attention to this fact, that constructing a magnitude through multiplication is really a completely different process from destroying a magnitude through division. — Metaphysician Undercover
I don't understand how you would build an irrational length segment. — Metaphysician Undercover
What the law of identity says is that a thing is the same as itself. This puts the identity of the thing within the thing itself, not as what we say about the thing, or even the name we give it — Metaphysician Undercover
First, to be a thing is to have an identity — Metaphysician Undercover
Second, a thing is unique, and no two things are alike, and this is the principle Leibniz draws on — Metaphysician Undercover
So the law of identity is not concerned with how we refer to objects, it is a statement concerning the real existence of objects, as the objects that they are, independent of what we say about them — Metaphysician Undercover
OK, but I don't understand how all this can be related to irrational numbers. — Mephist
Division between integers is repeated subtraction ( A/B you count how many times you have to subtract
B from A to reach 0 ); multiplication between integers is repeated addition ( A*B you add A B times starting from 0 ). — Mephist
The definitions are quite symmetric between each-other. What do you mean by "division presupposes no such base units"? OK, A/B is not an integer ( there is a reminder ) if A is not a multiple of B. Again: what does this have to do with physical space-time? — Mephist
By using compass and straightedge (as described by Euclides) you can build all the lengths that can be obtained from the integers with a finite sequence of operations of addition, subtraction, multiplication, division, and taking square roots (https://en.wikipedia.org/wiki/Straightedge_and_compass_construction). Square roots are not so special from this point of view. — Mephist
OK, I translate this as: you can always distinguish a thing (meaning: physical entity) from all the other things. Not quite true in quantum mechanics, but let's assume it is. — Mephist
OK, but when you give a name to a concrete object, the name is a reference that identifies always the same concrete object, isn't it? — Mephist
Anyway, my main objection to what you say is that you don't explain how to use the fact that square roots are irrational (some of them) to deduce something about physical space-time. A physical theory in my opinion (even if limited) should be falsifiable in some way (meaning: should be usable to predict that something should happen, or that something else can't happen). And if it's not physics but only mathematics, then there should be some kind of logical "proof". Don't you agree? — Mephist
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