I think your thesis "stick to finitism when teaching basic math" misses the obvious point of how incredibly messy and complex finitism is, both as a mathematical approach and as a practical application. The overwhelming majority of mathematical applications are based on the continuum - physics, engineering, etc. And as someone with your background knows perfectly well, and as you in fact emphasize in your post, when doing practical calculations, at some point you have to discretize those continuum models - which is not simple at all, especially if you want to do it robustly and accurately! In fact, you always want to keep them nice and continuous for as long as you can, and only discretize when all your analytical resources are exhausted, because once you do that, there's no going back.

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

You're a funny guy.

You: The moon is made of green cheese.

Me: Actually scientists think it's made of dirt and rocks and stuff.

You: Ok good, now that we agree the moon's made of green cheese ...

Man why you do me like this?

OK, so we agree that if so-called "mathematical objects" are things which can be measured, — Metaphysician Undercover

No. There are nonmeasurable sets. Some mathematical objects can be measured and some can't. I would never say that "a mathematical object is a thing that can be measured" since that's false, meaningless, and misleading. You just made it up and decided that I said it. You keep doing this. Why?

Euclidian geometry creates distances which cannot be measured by that system. — Metaphysician Undercover

I've told you half a dozen times already that:

* The length of the diagonal is the Euclidean distance between the points (0,0) and (1,1), which is $\sqrt 2$. We can define this via a metric, which is what I just did. We can also define it in terms of measure theory. I gave the links to those subjects earlier.

You're hung up on the infinite decimal business but I've explained to you repeatedly that:

* The fact that the decimal representation of a number is infinite, tells us nothing about whether the number itself is essentially a finitary or infinitary object. For example 1/3 = .333... has an infinite decimal expression but it can be summarized as "a decimal point followed by all threes." That's a recipe to produce a arbitrary number of decimal digits of 1/3. Likewise there is a recipe to product the decimal digits of $\sqrt 2$; as well as a recipe to produce the decimal digits of $\pi$.

* $\sqrt 2$ is computable; it has a finitely expressible continued fraction representation; and it lives in a finite extension of the rational numbers if one is an algebraist and doesn't like limits and infinite series. By these criteria, $\sqrt 2$ is a finitary object.

* 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.

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

LOL. My wonderment is that you consistently fail to engage with anything I say; repeatedly claim I said the opposite of what I actually said; and stubbornly cling to your misunderstood fractured math from high school.

You say "such a thing" as if $\sqrt 2$ is beyond the pale, whereas rational numbers are wonderful. You just made this up. Both classes of numbers are equally fake or equally real, depending on how you look at it. You don't want to engage with this point, go in peace then. I can't do any more for you.

That geometrical system is causing us problems, inability to measure things, by creating distances which it cannot measure. — Metaphysician Undercover

You keep repeating this without engaging with the fact that math says you're wrong. You keep repeating this over and over and over. I can't say anything beyond what I've already said many times.

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

I would agree with that; except that utility is not the ONLY consideration. For pure mathematians, beauty and interestingness have higher virtue than utilility. Utility is for the physicists, and we know what THEY do with mathematics!! [They mangle the hell out if it for their own nefarious purposes].

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

Yes, in general. But in this particular case, what you think is a defect is not. You're hung up on infinite decimals, but infinite decimals don't tell you anything about whether a number is rational or not. 1/2 = .5 = .49999.... It has TWO distinct decimal representions. That tells us nothing about the real number 1/2. It just tells us that decimal representation is a little buggy. Continued fractions are better. Turing machines are better. Infinite series representations are better.

Bad is not necessarily the opposite of good, because these two may be determined according to different criteria. — Metaphysician Undercover

Yes all these generalities are wonderful but they're in service of a point that is wrong. Since $\sqrt 2$ is a finitary object, your general point doesn't apply here. There's nothing wrong with $\sqrt 2$ except your psychological block about it. Was a screechy math teacher mean to you? I can relate. I still can't do high school trig for shit because of my screechy trig teacher. She set my math development back years.

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

For Christ's sake, knock it off with this point. You're absolutely wrong.

, can we say that it is bad, despite the fact that it is good in many ways? — Metaphysician Undercover

No.

How should we proceed to rid ourselves of this badness? — Metaphysician Undercover

Who will rid me of this meddlesome priest!!

Dude there is no badness. You had a bad high school math education -- not your fault, I'm sick at heart at the state of public math education -- but you refuse to move past it. You're just wrong on the facts.

Should we produce another system, designed to measure these distances, which would necessarily be incompatible with the first system? — Metaphysician Undercover

Please stop. You were wrong the first time, you're wrong the hundredth time.

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

You are so full of yourself you won't stop to engage with the points I (and others) have made to correct your misunderstandings. $\sqrt 2$ is a finitary object. It only requires a finite amount of information to completely specify its decimal digits. Why won't you acknowlege this fact?

Come on, get real fishfry. Check Wikipedia on set theory, the first sentence states that it deals with collections of "objects". — Metaphysician Undercover

I am not responsible for what people type into Wikipedia. Some math articles are very good, some are highly misleading.

In ZFC there is nothing called an object. There are only sets; and sets are an undefined term. ZFC consists of a collection of axioms about how an undefined operator called $\in$ behaves. You can verify this by checking any university or graduate text on set theory.

Once again you are giving the high school definition. It's confusing you.

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

Nonsense. Set theory precedes objects. We use set theory to construct numbers, functions, matrices, topological spaces, and all the other "objects" of mathematics. An object literally is some gadget we construct out of sets. And sets are undefined. Nobody knows what a set is. We have private intuitions, but set theory itself supports no preferred interpretation.

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.

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

I'm explaining to you what sets are, from the point of view of the mathematical discipline of set theory. You don't want to get it, fine by me. And you're right, we're pretty much at a point of completion here. I've made every point I have to make at least half a dozen times. I'm happy to abandon this thread.

When I started, because of your arrogance and certainty and wordiness, I thought perhaps that I was missing some subtle philosophical viewpoint.

Instead it turns out that you're just stuck on some psychological discomfort with what you learned (badly, and again not your fault) about the square root of 2 and its decimal representation.

Having satisfied myself that I'm not missing some subtle point of philosophy; I have turned my efforts to trying to educate you about mathematics. You don't seem to be receptive and now I'm just annoying you. So I'll happily withdraw from the conversation. I stand by everything I've written.

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

Incredible. I went to great lengths to explain to you that mathematical equality is an expression of the law of identity. That's what my Peano axiom proof that 2 + 2 = 4 was all about. You didn't even engage.

Once again you have imputed to me a position that is the direct opposite of the one I expressed.

You have not explained how acceptance of a mathematical tool, through convention, converts it from a funny gadget, to an object. — Metaphysician Undercover

I did. The passage of time. As the great physicist Max Planck said: science advances one funeral at a time. What he meant was that the old experts are not convinced by the new methods. Rather, the old guys die off and are replaced by a new generation that has grown up with the new ideas. That's how we came to accept rational numbers in the first place, and then irrationals.

If you cannot demonstrate this conversion, then either the tool is always an object, or never an object. — Metaphysician Undercover

I already did, at least three times. Are you denying history? Read up on the history of negative numbers, zero, rational numbers, real numbers, complex numbers. I keep explaining this to you and you keep avoiding engaging with the point.

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

You're so tied up in words you can't think straight.

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 truth is what you learned (badly) in high school or Wikipedia. Anything else is a lie. Whatever dude.

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

There is no technical term called an object in set theory. It's something you made up. Sets don't contain "objects." They contain only other sets, if they contain anything at all.

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

You're embarrassing yourself. Your mathematical philosophy is unsophisticated because your knowledge of math is nil. You are psychologically stuck to your wrong ideas and you're incapable of engaging substantively with any point that anyone makes.

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

Whateva whateva. I hope I'm getting to the end of this soon. This is my last post to you. I'm a fool if I continue to engage.

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

You 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.

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

That doesn't even make any sense. It's a collection of words that seems to convey a coherent argument about something but simply doesn't.

So now you don't believe in 1/3? I think you just refuted yourself.

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

I'm out of steam.

This demonstrates that there is a problem we have with dividing magnitudes, which has not yet been resolved. — Metaphysician Undercover

Dividing magnitudes. So now you don't believe in rational numbers either. We are making progress. That's right! Rational numbers are just as fictional or just as real as irrational numbers.

Could understanding be dawning?

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

There's nothing bad to cover up. The 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.

I 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.

Peace, friend.

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

K-theory, Category theory, etc. might enforce this view. I remember years ago hearing a well-known algebraist joke that, "K-theorists will tell you, "All you have to do is believe me and I can prove it!'." However, moving up into more abstract or general levels with new definitions and relationships, while simplifying certain aspects of math below those levels, may or may not solve complicated problems at lower levels. For example, "Soft" analysis doesn't solve all the problems "Hard" analysis presents. I am well aware of this having done research in the latter. On the other hand, moving higher up, greater generality, in a subject can be wonderfully rewarding, and it certainly provides avenues of imaginative research for grad students. The lower level stuff has frequently been "mined out" and what remains can quite difficult.

However, this is a side track, unimportant in this thread. :nerd:

Well, I am surprised. I didn't expect somebody to agree with that kind of categorical assertions! :razz:
I mean: it's clear that finding the right definitions it's not all. And it's extremely conceited to say "I'll tell you what the whole mathematics is about!". But there's something true in what I wrote, and I wanted to see if somebody agrees without spending too much time to explain what I mean :smile:

However, re-reading that thread, I see that I threw even harder (and even less comprehensible) stuff, like this one: "A formal proof makes only use of the computational (or topological) part of the model. The part that remains not expressed in formal logic is usually expressed in words, and is often related to less fundamental parts of physics, such as, for example, the geometry of space.".
I guess nobody replied to this one because everybody thought that it doesn't make sense at all :joke:

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

No, I most definitely would not want that, and I've already explained why. I don't think it's a good idea to do a second bad thing to cover up an original bad thing. So you'd have to demonstrate to me first that the original thing which I consider to be bad (irrational numbers), is not really bad, with reference to solid ontological principles, rather than referring to what I called the second bad, which is just a cover up of the first bad. I have no inclination to learn the cover up, call it a psychological condition if that makes you happy.

You keep repeating this without engaging with the fact that math says you're wrong. — fishfry

I am arguing against accepted mathematical principles. How is "math says you're wrong" any sort of a counter argument? Of course math says I'm wrong, that's a given.

I am not responsible for what people type into Wikipedia. Some math articles are very good, some are highly misleading. — fishfry

OK, math says I'm wrong, but Wikipedia says you're wrong. Now we're even, both wrong.

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

Fishfry! Get with the program, wake up and smell the coffee! We've engaged in these discussions for weeks now, you know it's pointless to speak sophisticated math at me. You're wasting your time, we're discussing philosophy on this forum, not sophistic math.

I did. The passage of time. — fishfry

Ok, so as time passes, a "funny gadget" is magically converted into a "mathematical object". Tell me another one.

It's something you made up. — fishfry

No, I got it from Wikipedia, someone else made it up. But how is that any different from your "funny gadgets", which someone makes up, and through the passage of time magically turn into mathematical objects?

You are psychologically stuck to your wrong ideas and you're incapable of engaging substantively with any point that anyone makes. — fishfry

Actually, it's you who has not engaged in any of the substance of my post, and has regressed to ad hom, and repeated insistence of "your wrong".

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

Huh, I don't see any evidence of that scrutiny, only repeated assertions, "you're wrong", "you're wrong", you're wrong".

That's right! Rational numbers are just as fictional or just as real as irrational numbers. — fishfry

Right, we've been through this already they are fictions, like your "funny gadget". But in logic we look for consistency, along with the capacity to fulfil the purpose. Why would a geometrical system produce a distance which is impossible to measure? How is this consistent with the purpose of geometry?

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

I thought we were talking about fictions. How is truth relevant?

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

I never claimed to know a lot about philosophy of math. I didn't even know there is such a thing. I've been arguing ontology. No wonder we're on different pages.

Fishfry! Get with the program, wake up and smell the coffee! — Metaphysician Undercover

I love coffee. My favorite part of the day is my morning coffee. I can't wait to wake up tomorrow morning and smell the coffee. Often I grind my fresh artisinal beans and then bring the container to my nose, inhaling the aroma. Ah, coffee. Nectar of the Gods.

I hope you will not mind too much if I refrain from commenting on other topics. If I said anything it could only be what I've said before. Little would be gained in further punishing the keys of my laptop.

it's pointless to speak sophisticated math at me. — Metaphysician Undercover

Ignorance as a debating point. "Your argument stands refuted because I'm incapable of understanding it." I can't top that.

My favorite part of the day is my morning coffee. — fishfry

Seems we have something in common.

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.

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

I could respond but what would be the point? It is a logical truth that IF you believe in the rational numbers then you must necessarily believe in the rational numbers augmented by the square root of 2. It's a simple logical procedure to go from one to the other. You want to complain that this is a sophisticated mathematical argument. Actually it is. But I've just explained it in a most understandable way. You needn't follow the details of the procedure. What matters is that there is one. You can literally build a square root of 2 out of the rational numbers. I have in fact outlined the procedure a couple of times already.

I can't expect you to follow mathematical arguments. I am simply making you aware of the existence of these arguments.

Your preference not to engage with mathematical arguments does not give you the right to deny that such arguments exist. You don't have to follow the algebraic details. You do have to understand that from the standpoint of pure logic, the correctness of the rationals implies the correctness of a number system that includes the rational augmented with the square root of 2.

Else you really are trying to use ignorance as a weapon. "I don't understand it so don't waste your time explaining it to me," is acceptable if lame. But "I don't understand it therefore it's false," I hope you can see is not a sensible argument at all.

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. How is that any different? Without reaching the end, you have no definite solution, an approximation of something unresolved.

The construction procedure you described is never ending, just like the never ending digits. — Metaphysician Undercover

No, you're thinking of something else. I'm talking about the algebraic construction. Which I'll outline.

* Say you believe in the rationals. You believe in the rationals? Ok.

* Now arbitrarily make up a symbol, call it $\sqrt 2$, but bear in mind that this is an entirely arbitrary symbol that has no meaning. It's just some squiggles I type in.

* Now consider the set of all formal expressions $a + b \sqrt 2$. Again, these are just marks on paper. They have no meaning.

* We can define "addition" on these expressions componentwise. So

$(a + b \sqrt 2) + (c + d \sqrt 2) = (a + c) + (b + d) \sqrt 2$

* Likewise we can define the "product" of two such expressions using the everyday distributive laws. FOIL if you learned that awful acronym designed to replace understanding with mindless drudgery. God I hate what passes for math education. My friend @Meta your mathematical ignorance is not your fault. I blame your teachers and the textbook committees and the educrats of your high school years.

* Now the set $\{a + b \sqrt 2 : a, b \in \mathbb Q \}$ happens to have a mathematical structure identical to that of the rationals; and in addition, it contains a square root of 2.

* If one then objects that these are "only formal symbols," well after all what are rational numbers but formal symbols that obey rules? And in fact we can go further and construct, out of bits and pieces of set theory, a mathematical structure that is the set-theoretic implementation of this set of symbols.

Fact: If you believe in the rationals, you must believe in the rationals augmented by the square root of 2.

You want to make some kind of distinction that the "rationals are actual" in some sense. But they're not. They're just as fictional.

ps -- You make a mathematical claim, "Sqrt 2 doesn't exist." Then you reject any mathematical counterargument. You can't lose that way, but you can't convince anyone else.

If you are going to make a mathematical claim you need to be able to accept a mathematical disagreement.

The square root of two just won a Golden Globe award!

:gasp:

The square root of two just won a Golden Globe award! — jgill

LOL

* 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

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?

Fact: If you believe in the rationals, you must believe in the rationals augmented by the square root of 2. — fishfry

Why do you believe this? "The square root of two" has no valid meaning in the rational number system. This means that taking a square root is not a valid operation. So your claim is like saying if you believe in the rationals then you believe in the rationals augmented by a tree. You can't augment a system by something which is inconsistent with the system, that creates a contradiction, or at best, meaninglessness.

Square roots are a problem in mathematics, as is demonstrated by "imaginary numbers". At first glance, it appears like a square root is simply the inversion of the power of two. But the power of two is a valid procedure whereas the square root is not. If we define "square root" as the inversion of the power of two, then we'll find many numbers which simply do not have a square root. Why not accept this as a natural fact of numbers, rather than trying to force a square root onto these numbers?

If you are going to make a mathematical claim you need to be able to accept a mathematical disagreement. — fishfry

So long as you spell out your premises, and they are acceptable, I'm good to go. But it's already been proven that the square root of two is not a rational number. Why flog a dead horse? Why not move on, and inquiry what this principle tells us about numbers and spatial relations, instead of trying to disprove it.

Why not move on, and inquiry what this principle tells us about numbers and spatial relations, instead of trying to disprove it. — Metaphysician Undercover

I am curious to know: do you have an answer to this question?

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

You want to make a mathematical claim (sqrt 2 doesn't exist) but you won't accept a mathematical response. Makes for pointless conversation.

I don't accept set theory on the basis that it violates the law of identity — Metaphysician Undercover

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.

This means that taking a square root is not a valid operation — Metaphysician Undercover

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. :smirk:

I am curious to know: do you have an answer to this question? — Mephist

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.

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 amore complex example of this simple problem of division, the issue of the remainder.

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

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. My argument is that the problem is fundamentally an ontological problem, and the objective ought to be to resolve the problem with principles which are ontologically sound.

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

As I demonstrated in the other thread, the "identity" expressed here as "equals", is not consistent with "identity" as expressed by the law of identity. Therefore despite your claim that set theory is based in "identity", it uses a form of identity which is in violation of the law of identity.

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.

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

A valid operation is one carried out with consistency according to consistent laws of a system. The example of imaginary numbers, as well as the various different attempts to prove the square root of two, demonstrate that there is a lack of consistency to the square root procedure.

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

At the time, I responded thoughtfully to your ideas. You never once engaged with the points I made. Now weeks later you're still repeating your claims without ever having responded to the points I made. It's not productive to engage with you.

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

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 (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?

If this is the idea, I think the problem with this kind of physical theory is that all laws of physics are expressed in terms of differential equations (even the ones that describe "quantized" entities), and if quantum mechanics is right, it doesn't even make much sense to speak about an exactly determined physical length: physical space appears to be much more weird than a simple 3-dimensional geometric structure.

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

You mean that there is no defined physical procedure to divide a generic geometrical segment by another? 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). That's in Euclides' elements. Can't be this counted as division? If not, what do you mean by "law of division"?

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

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?

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

No, that does not follow. The irrationality of $\sqrt 2$ is a purely mathematical fact. It tells us nothing about the physical world.

But I can use numbers to describe (or model) physical processes (experiments):

1. Call Build_Side(N) the physical process of putting N sticks in line one after the other, along the side of a square. N is a natural number (abstract mathematical object), but the process of putting N sticks in line is a real, physical experiment.

2. Call Build_Diagonal(M) the physical process of putting M sticks in line one after the other, along the diagonal of the same square.

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, and the physical process of trying to build the square withe the sticks cannot be realized. ( well, OK, you have to build two sides of the square and the diagonal at the same time, but you get the what's the point! )

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

All physical measurement is approximate. You can't have a physical stick of length 1. It's not only impossible, it's meaningless. There is no physical apparatus in the world, even in theory, that could do any better than to say that "The length of the stick is 1 +/- .00005 with 99.343% certainty. I'm making up the numbers but that is what the nature of physical measurement is: a number, an error tolerance, and a probability that the true value is within the tolerance.

In the real world you can't measure the diagonal and you can't measure the sides. You can't measure anything with absolute precision. In classical physics, you can't but God can. In quantum physics, even God can't. To clarify that: in classical physics, we can't possibly measure the exact length of a stick, but at least in theory the stick does have a specific length. In quantum physics, a stick has no length at all until we measure it; at which point, the classical problem of the inexactness of physical measurement kicks in.

Well, in the current theory of the physical world (standard model, or whatever variant of it you prefer) all atoms of the same element are supposed to be EXACTLY the same (indistinguishable, even in principle, with absolute precision), right?

You are right, we will never be able to check if this theory is correct with absolute precision, not even in principle, because all physical measurements must necessarily have a limited precision.
Nevertheless, in principle (if you have enough computing power and the model is complete and consistent - I know, that's a big if) you can use the mathematical model to make predictions about the result of experiments with arbitrary precision.

So, in a model of the physical world where all distances have to be multiple of a given fixed length (I don't know if such a model exists, but let's assume this as an hypothesis), there cannot be squares
made of unit lengths. I don't know what these unit lengths are made of: they are simply the building blocks of my model, the same as the "strings" of string theory or the "material points" of Newtonian mechanics!

By the way, to be clear, I don't believe in this theory! :smile:

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

I do not think that "there cannot exist any fundamental minimal length of physical space" is a reasonable starting point. If space has physical existence, then it has limitations just like any other physical things. So we ought to assume that space must have some fundamental "shapes" just like you suggest. 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. 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.

You mean that there is no defined physical procedure to divide a generic geometrical segment by another? — Mephist

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.

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

I don't understand how you would build an irrational length segment.

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

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. This is a fundamental ontological statement about what it means to be a thing. First, to be a thing is to have an identity (but this is irrelevant to the identity we give the thing, it is the identity that the things has by virtue of being the thing that it is). Second, a thing is unique, and no two things are alike, and this is the principle Leibniz draws on. 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.

Once it was believed that space is an aether, — Metaphysician Undercover

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.

Were you merely wrong or mistaken, not an issue. We're all wrong or mistaken at least sometimes. But your errors run deeper, and beyond sport, if that's what you're about.

This, for example. Are you serious?

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

And,

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

You never thought I ever had it, that's why you'd tell things like get back on your meds. Is the fact that you keep addressing my post with such nonsense evidence that you're losing it?

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

OK, but I don't understand how all this can be related to irrational numbers.

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

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 ).
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?

I don't understand how you would build an irrational length segment. — Metaphysician Undercover

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.

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

Hmmm... :worry: maybe...

First, to be a thing is to have an identity — Metaphysician Undercover

OK

Second, a thing is unique, and no two things are alike, and this is the principle Leibniz draws on — Metaphysician Undercover

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.

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 when you give a name to a concrete object, the name is a reference that identifies always the same concrete object, isn't it?

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?

OK, but I don't understand how all this can be related to irrational numbers. — Mephist

The problem of irrational numbers arose from the construction of spatial figures. That indicates a problem with our understanding of the nature of spatial extension. So I suggested a more "real" way of looking at spatial extension, one which incorporates activity, therefore time, into spatial representations. Consider that Einsteinian relativity is already inconsistent with Euclidian geometry. If parallel lines are not really "parallel lines", then a right angle is not really a "right angle", and the square root of two is simply a faulty concept.

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

Ok, we can look at division as a matter of asking how many times we can subtracting B from A, as you say. The issue is that in many cases one does not reach 0, and this is what we call the remainder. So the problem is, how do we deal with the remainder. If we are dividing ten by three, we get a remainder of one. In this case, you might divide the unit into three. But in most practical circumstances, if you were dividing a group of objects, it would be unfeasible to split up one of the objects, rendering it useless. So the remainder is very often a problem in division.

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

No, division and multiplication are not at all symmetrical, because you never have a remainder in multiplication. In multiplication, you take a designated number as the "base unit", a designated number of times, and you never end up with a remainder. You have no such "base unit" in division, you have a large unit which you are trying to divide down to determine the base unit, but you often end up with a remainder.

Evidence of this difference is the existence of prime numbers. These are numbers which we cannot produce through multiplication. We can still divide them, knowing there will be a remainder, but that doesn't matter, because there's often a remainder when we divide, even if the dividend is not prime.

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

On paper you produce "a representation" of the Euclidean ideals. That representation is something completely different from the square root, which is part of the formula behind the representation which you draw on paper. When I want to lay out a square corner, a right angle, on the ground, I might use a 3,4,5, triangle. In this exercise I am not using a square root at all. I could make this square corner without even knowing the Pythagorean theorem, just knowing the lengths of 3,4,5. But if one side of the right angle is to be 5, and the other side 6, I'll need to know the Pythagorean theorem, and then figure the diagonal as the square root of 61 if I am going to make my right angle.

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

That's not quite right. We, as human beings, cannot necessarily distinguish two distinct things, due to our limited capacities of perception and apprehension. So it's not quite right to say that you can always distinguish a thing from all other things. A thing is distinct from other things, but we cannot necessarily distinguish it as such. And that difference may be a factor in quantum mechanics.

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

Right, but to perceive a thing, name it "X", and then claim that it has the "identity" of X, is to use "identity" in a way inconsistent with the law of identity. You are saying that the thing's identity is X, when the law of identity says that a thing's identity is itself, not the name we give it. The law says a thing is the same as itself, not that it is the same as its name.

Consider that human beings are sometimes mistaken, so it is incorrect to say "the name is a reference that identifies always the same concrete object". The meaning of the name is dependent on the use, so when someone mistakenly identifies an object as "X", when it isn't the same object which was originally named "X", then the name doesn't always identify the same concrete object. And, there are numerous other types of mistakes and acts of deception which human beings do, which demonstrate that the name really doesn't always identify the same concrete object, even when we believe that it does.

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

Do you recognize that Einstein's relativity is inconsistent with Euclidian geometry? Parallel lines, and right angles do not provide us with spatial representations that are consistent with what we now know about space, when understood as coexisting with time. My claim is that the fact that the square root of two is irrational is an indication that the way we apply numbers toward measuring space is fundamentally flawed. I think we need to start from the bottom and refigure the whole mathematical structure.

Consider that any number represents a discrete unit, value, or some such thing, and it's discrete because a different number represents a different value. On the other hand, we always wanted to represent space as continuous, so this presents us with infinite numbers between any two (rational) numbers. This is the same problem Aristotle demonstrated as the difference between being and becoming. If we represent "what is" as a described state, and later "what is" is something different, changed, then we need to account for the change (becoming), which happened between these two states. If we describe another, different state, between these original two, then we have to account for what happens between those states, and so on. If we try to describe change in this way we have an infinite regress, in the very same way that there is an infinite number of numbers between two numbers.

If modern (quantum) physics demonstrates to us that spatial existence consists of discrete units, then we ought to rid ourselves of the continuous spatial representations. This will allow compatibility between the number system and the spatial representation. Then we can proceed to analyze the further problem, the change, becoming, which happens between the discrete units of spatial existence; this is the continuity which appears to be incompatible with the numerical system.





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