premises of my argument that God can't exist! — TheMadFool

Maybe God put all these ideas in your head. I can never understand the mindset of people who use logic to talk about the existence of God. God exists outside of the bounds of logic. God is a matter of faith, not logic. One believes or not. Or one has experienced miracles or not. But of course we all experience miracles every day, being alive, breathing the air. All the better if we have a roof over our head and we know where our next meal is coming from. If God didn't provide that, I'm a damned lucky fool. I don't believe in an anthropomorphised God, but I don't believe in an entirely mindless universe either. I'm an agnostic: "a person who believes that nothing is known or can be known of the existence or nature of God or of anything beyond material phenomena; a person who claims neither faith nor disbelief in God." That fits me to a T.

I mean no disrespect to constructive mathematics. — jgill

I disrespect constructive math all the time. Actually over the past couple of years a handful of constructivists showed up here and I had interesting conversations with them, but always ended up baffled and they left. I hope it wasn't something I said, I'm just trying to learn. These were generally people coming to the subject from the computer science side and never the math side, so they had no appreciation for standard math. Made for frustrating conversations both ways.

Thanks for opening my eyes a bit to formalized set theory and for your patience! — jgill

Glad to help.

Note the underline bit, "...larger than any real or natural number". Sounds remarkably similar to "...no greater can be conceived" the main premise in the ontological argument. — TheMadFool

Didn't read any more of the thread than this, only jumping in with a mathematical correction. In math there are quantities that are larger than any finite number, but way smaller than other transfinite numbers. For example the cardinal $\aleph_0$ is greater than every finite number, yet smaller than $\aleph_1$, which is smaller than $\aleph_{\text{googolplex}}$. I realize this is primarily a religious thread and not a mathematical one so forgive the interruption. But your statement is no longer "remarkably similar" to "no greater can be conceived," as of the 1870's. Georg Cantor was the first human to conceive of, and logically prove the mathematical existence of, an endless hierarchy of larger and larger infinities. Infinity, at least mathematical infinity, is no longer just one single thing that's "larger than any finite number." It's an entire complex world of interrelated ideas of ordinal and cardinal infinities, respectively expressing order and quantity.

Many humans, mostly math majors, easily conceive of these things once they've had the proper training. The theologians should pay attention. Cantor himself was a very religious man, and believed that after his endless hierarchy of infinities, the ultimate infinity was God. He called it the Absolute infinite, and denoted it $\Omega$. Cantor's mathematics is universally accepted now, while his theological ideas are forgotten by everyone except historians. I imagine he'd be disappointed by that.

Just sayin'. I'll leave you learned theologians alone now, I don't know anything about the subject. Well I know one bit, when I think of the ontological argument I know about William Lane Craig and his misuse of set theory to prove the existence of the Christian God. So I haven't got a very good impression of theological arguments involving infinity.

The connection is the fact that the axiom of choice is equivalent to the law of excluded middle, which for infinite objects dissociates truth from derivation. This in itself wouldn't imply platonism if it wasn't for the fact that most proponents of classical logic and ZFC make no attempt to justify the formalisms pragmatically with respect to real world application. — sime

I do not believe you have correctly stated Diaconescu's theorem. (Didn't think I knew that one, did you!) I quote from Wiki:

In mathematical logic, Diaconescu's theorem, or the Goodman–Myhill theorem, states that the full axiom of choice is sufficient to derive the law of the excluded middle, or restricted forms of it, in constructive set theory.

So AT BEST, AC implies LEM. And NOT the converse. So your claim of equivalence is not supported by Wikipedia. I have to trust Wiki on this point because I haven't studied this much, just remember hearing about it.

Secondly, the qualifier, "in constructive set theory." Does that mean that the implication ONLY holds in constructive set theory? Or is that an ambiguous statement on the part of some Wiki author. I don't know. But you've gone way too far in your claim of equivalence.

Moreover, you have not even remotely justified your claim that AC is equivalent to Platonism. You wrote a sentence that I could neither parse nor understand, and I don't think it's true. I'm talking about, "This in itself wouldn't imply platonism if it wasn't for the fact that most proponents of classical logic and ZFC make no attempt to justify the formalisms pragmatically with respect to real world application." Can you rephrase that so I can understand what you're saying?

(edit) I'm beginning to unpack a little of that. "Most proponents of classical logic and ZFC make no attempt to justify the formalisms pragmatically." Why on earth should they? That's not their job. And that's not even Platonism. Platonism doesn't say that the abstract math thingies refer to the real world. It says that the abstract math thingies exist in some nonphysical Platonic world. So even here I don't think you're using the technical terms correctly.

Also, "... which for infinite objects dissociates truth from derivation. " What? if you reject LEM then truth = derivation? That's false. Derivation is syntactic, truth is what's true in some model. That doesn't change just because you reject LEM.

Yes, I'm already aware of all of that, and was only speaking approximately on set theory. My point was only attacking the idea that quantity is reducible to ordering. — sime

My most humble apologies, I do not know what this is in reference to. Quantity and order are surely different, that's why there are cardinals and ordinals.

choice principles and LEM are approximately equivalent — sime

"Approximately equivalent?" What does that mean? Are 12 and 14 approximately equivalent? Apples and rutabagas? I don't think you are making your point. And to be fair I don't know much about constructive math (and not for lack of trying), so I assume you have something interesting to say here, but you're not saying it clearly.

I looked at that article briefly. I did not see mention of an "approximate" equivalence. — TonesInDeepFreeze

Glad you said that, I thought I was being too snarky jumping on that phrase.

"And of course, we know that LEM does not imply AC, since we know that ZF is consistent with ¬AC while LEM holds." (MathStackExchange) — jgill

@jgill This place is corrupting you! You're supposed to say that as a complex variables guy you don't think much about foundations! This was most impressive.

This stuff just gets more bewildering as time goes on. :worry:

(One reason math has become so abstract is that classical areas of investigation have been "mined out". Professors need suggestions for research topics for their PhD students. So, create new definitions and/or generalize.) — jgill

Constructive math is making a comeback these days because of the influence of computers. There are computerized proof assistants, and they tend to operate on some form of constructive logic or intuitionistic type theory. Brouwer was the great 1930's intuitionist, so I thin of all this as "Brouwer's revenge."

deleted post — GrandMinnow

Would that others show so much circumspection. Myself included from time to time.

Hold on a minute! — TheMadFool

Ok ...

That's precisely the point. You can't tell the difference between randomness and determinism (sorry, I can't find a better word). If so, of what use is the distinction? I phrase I learned in an introduction to philosophy book seems apt: A distinction without a difference. — TheMadFool

We can know for certain that a pseudorandom sequence that we generated from an algorithm is not actually random.

But given an apparently random sequence, we can't know for sure if it's "truly" random (if that concept is even meaningful or possible), or merely pseudorandom.

So the point of pseudorandomness is that it works one way. We can know for sure that a p-random sequence is not actually random. But given an apparently random sequence, we can't tell if it's truly random or only p-random.

But I don't follow your point. Anything that comes out of a deterministic process is not random, by definition. So a p-random sequence is not random. That's a meaningful statement, I don't know why you don't think so.

When the typical person says something is random, they mean it's "truly random," ie NOT the output of a deterministic process. If it's deterministic, it's not random, even if it looks random.

Excellent commentary. Sitting in the bleachers awaiting MU's reply. — jgill

I'll join you for a hot dog and a beer.

The issue I brought up with fishfry is the distinction between representation and imagination. Fishfry allows that "abstraction" might encompass both of these, such that imaginary ideas could be a useful part of a representative model. — Metaphysician Undercover

I was just reading about the Frege-Hilbert dispute. As I understand it, Hilbert was saying that axioms are formal things and it doesn't matter what they stand for as long as we can talk about their logical relationships such as consistency. Frege thought that the axioms are supposed to represent real things. I'm not sure if I'm summarizing this correctly but this is the feeling I got when I was reading it. That I'm taking Hilbert's side, saying that the axioms don't mean anything at all; and you are with Frege, saying that the axioms must mean whatever they are intended to mean and nothing else.

https://plato.stanford.edu/entries/frege-hilbert/

Is this what you're getting at?

You then complained about my map analogy, but I hope you will agree that you could object to my analogy without necessarily refuting my thesis. If you don't like maps, forget the maps.

ps -- I found another nice article.

https://academic.oup.com/philmat/article/13/1/61/1569375

You can skip the category theory stuff, scroll down to here:

During the second half of the nineteenth century, through a process still awaiting explanation, the community of geometers reached the conclusion that all geometries were here to stay … [T]his had all the appearance of being the first time that a community of scientists had agreed to accept in a not-merely-provisory way all the members of a set of mutually inconsistent theories about a certain domain … It was now up to philosophers … to make epistemological sense of the mathe‐maticians’ attitude toward geometry … The challenge was a difficult test for philosophers, a test which (sad to say) they all failed …

For decades professional philosophers had remained largely unmoved by the new developments, watching them from afar or not at all … As the trend toward formalism became stronger and more definite, however, some philosophers concluded that the noble science of geometry was taking too harsh a beating from its practitioners. Perhaps it was time to take a stand on their behalf. In 1899, philosophy and geometry finally stood in eyeball-to-eyeball confrontation. The issue was to determine what, exactly, was going on in the new geometry.

I take this to be a reference to the fact that geometers studied various models of geometry, such as Euclidean and non-Euclidean, and were no longer concerned with which was "true," but rather only that each model was individually consistent. And philosophers, who said math was supposed to be about truth, were not happy.

And:

What was going on, I believe, was that geometry was becoming less the science of space or space-time, and more the formal study of certain structures. Issues concerning the proper application of geometry to physics were being separated from the status of pure geometry, the branch of mathematics.1Hilbert's Grundlagen der Geometrie [1899] represents the culmination of this development, delivering a death blow to a role for intuition or perception in the practice of geometry. Although intuition or observation may be the source of axioms, it plays no role in the actual pursuit of the subject.

(my emphasis).

I think that last bolded part is the heart of our discussion. I'm saying that we can study aspects of the world by creating formal abstractions that, by design, have nothing much to do with the world; and that are studied formally, by manipulation of symbols. And that we use this process to then get insight about the world.

And you say, how can these meaningless abstractions possibly tell us anything about the world? They're meaningless, they're not true. You are taking the Fregean position, that if the axioms are not about the world, they're nonsense.

Here's Paul Bernays explaining Hilbert's point of view:

A main feature of Hilbert's axiomatization of geometry is that the axiomatic method is presented and practiced in the spirit of the abstract conception of mathematics that arose at the end of the nineteenth century and which has generally been adopted in modern mathematics. It consists in abstracting from the intuitive meaning of the terms … and in understanding the assertions (theorems) of the axiomatized theory in a hypothetical sense, that is, as holding true for any interpretation … for which the axioms are satisfied. Thus, an axiom system is regarded not as a system of statements about a subject matter but as a system of conditions for what might be called a relational structure … [On] this conception of axiomatics, … logical reasoning on the basis of the axioms is used not merely as a means of assisting intuition in the study of spatial figures; rather logical dependencies are considered for their own sake, and it is insisted that in reasoning we should rely only on those properties of a figure that either are explicitly assumed or follow logically from the assumptions and axioms.

(my emphasis again)

I do believe this is what our conversation is about. You're a Fregean and I'm with Hilbert.

What do you think? For my part these articles have given me some insight into your point of view. If collections in the real world have inherent order, what sense does it make to postulate sets that have no inherent order? The answer is Hilbert's side of the debate; which, for better or worse, is the prevalent view in modern math.

There's more. This is a great article.

At first, Frege had trouble with this orientation to mathematics. In a letter dated December 27, 1899, he lectured Hilbert on the nature of definitions and axioms.3 According to Frege, axioms should express truths; definitions should give the meanings and fix the denotations of terms.

(my emphasis)

This is EXACTLY what you are lecturing me about! But as SEP notes, Hilbert stopped replying to Frege in 1900. Just as I ultimately had to stop replying to you. If you and Frege don't get the method of abstraction, Hilbert and I can only spend so much time listening to your complaints. I found a great sense of familiarity in reading about the Frege-Hilbert debate.

a set is normally specified as a collection of things which satisfy a given predicate, — sime

I foresee big trouble with this definition or my name's not Gottlob Frege. Well my name's not Gottlob Frege, but Frege proposed the same definition. Then Russell came along and said, "Ok smart guy, take the predicate to be $x \notin x$ and see what happens. You get a contradiction. Busted!

https://en.wikipedia.org/wiki/Russell%27s_paradox

In fact a set is ... well nobody knows what a set is. A set is whatever satisfies the axioms of some particular set theory, of which there are many. A thing could be a set in one set theory and not in another.

What you gave is the definition used in "high school set theory," sometimes called naive set theory. But that definition fails as soon as you examine it closely.

Ironically, it is the the platonists who insist that every set must be "well-ordered" which is an assumption equivalent to the axiom of choice. — sime

News to me. You claim that being a Platonist ixs equivalent to believing in the axiom of choice? I'd take those two things to be totally independent of one another. You could be a Platonist or not, and pro-choice or not. I don't see the connection.

But for those who deny the axiom of choice, it is nevertheless meaningful to compare the "sizes" of different sets even if the determined sizes are not synonymous to counting elements. — sime

The problem is that absent the axiom of choice, there are infinite sets that are not comparable to each other. That is, there are cardinals $x$ and $y$ such that none of $x = y$ nor $x \lt y$ nor $x \gt y$ are true.

That's actually the purpose of the axiom of choice, to make infinite sets behave. Without choice there's an infinite set that's Dedekind-finite; that is, a set that's not bijectively equivalent to any natural number 0, 1, 2, 3, ..., but that is not bijectively equivalent to any of its proper subsets.

So in fact your statement is inaccurate. If you deny the axiom of choice, it is sometimes MEANINGLESS to compare the sizes of sets.

I'm honestly stupefied — aRealidealist

That, I believe.

It's a small point. But you initially made an absolute statement, and you then qualified it as a relative statement after I pointed out your error to you. So congrats on understanding what I said to you and putting in the correction. And non-congrats on pretending you thought of it yourself. You can have the last word.

Thus you inadvertently grant my definition of motion. — aRealidealist

Thus you inadvertently refute your own statement. You said, and this is a direct quote: "For motion can only be defined as a change of place or position,"

But since by that definition everything is in motion relative to something, your definition is true but useless, since I can't use it to determine whether a thing is moving or not. You acknowledged that motion can only be defined relative to some other thing, you added after the fact when I pointed it out to you.

Uh, yes, it does, precisely because, although everything can be said to be in motion or changing place/position, everything isn't, in the same context, moving in the same direction, e.g., one thing can be said to be moving to the left another & the other thing moving to the right of other (or, depending on the context, vice versa); thus distinguish between movements by direction. — aRealidealist

So you agree that all motion is relative. Something is moving only in relation to something else, and not in any absolute sense. That's a lot different than what you said initially. Can you see that? If I'm sitting on the couch, I'm motionless with respect to the couch, but moving at very high velocity relative to the galactic core.

So how can I know, using your criterion, whether I'm in motion? Or if everything is motion relative to something, of what use is your diagnostic criterion? Pick an object. It's in motion relative to something. So everything is in motion. Which happens to be true. Nothing in the universe is stationary.

For motion can only be defined as a change of place or position — aRealidealist

Relative to what? If I am sitting on my couch watching Youtube videos, am I moving? Well no, relative to my couch. No, relative to my street, my city, and the earth. Yes, relative to the sun and the stars. The earth is moving around the sun, the sun is wandering through the galaxy, the galaxy is rotating and also moving through space at tremendous velocity.

See for example this article that has the exact numbers for how fast we're moving through space. Even us couch potatoes.

So, am I moving? And if motion is defined as a change of place or position, can you identify anything in the universe that's not moving? Even the universe is, according to multiverse theory, floating around in a sea of bubble universes. If everything is in motion, then your definition doesn't distinguish anything in the universe. It's not that what you said is wrong; it's that everything in the universe is in motion. So this doesn't tell us anything interesting. That speeding race car? It's in motion. That 200 year old statue? It's in motion too. And not just because some wokesters pulled it down :-)

Gabriel's horn, honestly, is not really different, because what's taken as infinite, in that context, is the rotation about the horn's surface - in other words, the revolution about the horn's surface or the rotation about the axis is what's taken as infinite, & not the object itself per se (if you consider the matter closely, I think that you should see this to be the case). — aRealidealist

The surface area of Gabriel's horn is most definitely infinite. There was a long thread on this subject a while back. 17 pages worth.

https://thephilosophyforum.com/discussion/10263/the-paradox-of-gabriels-horn/p1

Randomness, patternlessness, can be generated with an algorithm. — TheMadFool

That's a married bachelor. Algorithms are deterministic. They cannot create randomness.

The person playing this game will see no pattern in the coin tosses. P(heads) = 50% = P(tails). In other words this person will think the outcomes (heads/tails) is random and for all intents and purposes if all possible outcomes (heads/tails) are equiprobable, it is considered random. However the deus deceptor used a simple algorithm to generate this randomness. — TheMadFool

This is precisely what pseudorandomness is: A deterministic sequence that passes all known statistical tests for randomness. It "looks random."

You don't need an evil demon, just a mathematical gadget called a linear congruential generator. That's how random() functions in programming languages work.

//ps hey are you familiar with this website https://www.cantorsparadise.com/ — Wayfarer

Heard of it, I'll have a look around. Thanks.

Can you unpack that a bit for the benefit of those without a background in mathematics? Say a bit about why it is the case, and what it means? Thanks. — Wayfarer

The story actually goes back to Leibniz, who "dreamt of building a machine that could manipulate symbols in order to determine the truth values of mathematical statements."

The problem got its modern form as the Entscheidungsproblem, which is German for "decision problem." According to Wiki,

The problem asks for an algorithm that considers, as input, a statement and answers "Yes" or "No" according to whether the statement is universally valid, i.e., valid in every structure satisfying the axioms.

By the completeness theorem of first-order logic, a statement is universally valid if and only if it can be deduced from the axioms, so the Entscheidungsproblem can also be viewed as asking for an algorithm to decide whether a given statement is provable from the axioms using the rules of logic. — Wikipedia

In 1936 Alan Turing published his famous paper, On Computable Numbers, With an Application to the Entscheidungsproblem. Turing was played by Benedict Cumberbatch in the highly fictionalized but definitely worthwhile movie, The Imitation Game.

In this paper Turing showed that there could be no mechanical procedure to decide the truth of mathematical propositions.

As part of his proof, he invented a formalized model of computation called the Turing machine (TM). A TM can be programmed to solve a given mathematical problem. He showed that there are many real numbers whose digits can be cranked out by a TM. He defined such a number as computable. For example any rational number is computable, since given two integers n and m we can just apply the grade school division algorithm to compute as many digits as we like of the quotient n/m.

Surprisingly, even many irrational numbers are computable. For example pi can be computed via the famous Leibniz series:

pi/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - ...

which can easily be turned into a computer program or TM.

However there are many noncomputable real numbers, whose decimal representation can never be computed by any TM or any computer program.

Now, how many real numbers are computable? Since each TM program consists of finitely many finite-length instructions, there are only countably many such programs: finitely many of length 1, finitely many of length 2, and so forth.

On the other hand we know from Cantor's diagonal argument and also from Cantor's theorem, that there are uncountably many real numbers. So there are "too many" real numbers to compute them all with TMs. These are the noncomputable real numbers. They are deserving of the name random, because there is no mechanical or deterministic process to generate their decimal representations.

Now when we say that "almost all" real numbers are noncomputable, we mean that a countable set is vastly smaller than an uncountable one. So "most" real numbers are noncomputable.

We can formalize this idea using measure theory. Any countable set has measure zero. So for example in the unit interval consisting of all the real numbers between 0 and 1, the set of computable reals has measure zero, and the noncomputable reals has measure 1.

Intuitively this means that if you put all the real numbers in a paper bag, closed your eyes, and reached into the bag and pulled out a real number, the probability is zero that it's computable, and probability one that it's not.

Another way to think of it is that if you flip a fair coin infinitely many times, regard heads as 1 and tails as 0, and put a binary point in front of the sequence, you have the binary representation of some real number. The probability that a computer program could crank out that exact sequence is zero; and the probability that no possible computer program or mechanized procedure could do so is one.

This is actually plausible. If we randomly flip infinitely many coins to generate a sequence of 0's and 1's, it's very unlikely that there will be any kind of pattern or mechanical procedure that predicts the sequence. It's far more likely that the sequence will be patternless, or random.

The noncomputable real numbers are sort of the "dark matter" of the real numbers. They don't have names. They don't have individual qualities or characterizations that uniquely describe any of them. They are simply there, holding the real line together.

So my thesis is that even beyond the mysteries of quantum mechanics, if anything in the world can legitimately be called random, it's the noncomputable real numbers.

Constructivist mathematicians don't believe in noncomputable numbers. They try to do all of math using only numbers that can be specified via algorithms or procedures. They're gaining some mindshare these days due to the influence of computers in math.

Well I wrote a lot but I left out a lot, so please ask if anything's not clear.

Congratulations! You've completed the OP — Wheatley

So you were only trolling? I don't get it.

'And what is psudorandom? — Wheatley

A sequence is pseudorandom when it's the output of a deterministic process, but the sequence satisfies all known statistical tests for randomness. When you call the random() function in any computer language, you're not getting a random number, but rather a pseudorandom one.

Pseudorandomness measures the extent to which a sequence of numbers, though produced by a completely deterministic and repeatable process, appear to be patternless.

https://en.wikipedia.org/wiki/Pseudorandomness

I'm not trained enough in mathematics to answer your questions. — Wheatley

A computable number is a real number whose decimal digits can be generated by a computer program.

Most real numbers are not computable. The digits of a noncomputable number are completely unpredictable. No program or recipe or algorithm or mechanical procedure of any kind can generate them. Noncomputable numbers are random if anything is.

You don't have to know much about this, only that most real numbers are truly random in any sense of the word you could name.

All irrational numbers are not truly random (theyre psudorandom). — Wheatley

Almost all reals are noncomputable. How is some noncomputable number pseudorandom but not random in your view?

Where can you find true randomnes? — Wheatley

Among the noncomputable numbers. Pick any one you like. Its binary digits are as random as anything can be.

"Idealized exactness" is not "truth". — Metaphysician Undercover

This is why it's not productive to continue this convo. I've spent the last couple of posts saying that math is a lie, math is fiction, math is untruth in the service of higher truth, and you put words in my mouth. It's not fun and there's no point.

The Pythagorean theorem is very true in the real world. — Metaphysician Undercover

Ok whatever. I have some stuff I'm dealing with in meatspace and maybe I'll get back to this tomorrow or the day after. But there's no point to this. I hope you will accept that. You claim I said things I've been saying the exact opposite of, and you take positions that I don't find sufficiently reasonable to interact with. There is no point to my replying. You do need to understand the concept of the necessary approximateness of all physical measurement. I can't imagine why you would take a stance so fundamentally wrong. You cannot draw a line of length 1 in the real world nor an angle of exactly 90 degrees. And you're the one who's convinced the square root of 2 doesn't exist, and now you say it not only exists, but you can draw it as the diagonal of an exact unit square. In order to have a conversation there has to be some small sliver of shared reality, and I find none here.

Like us animals have their own culture, they have their own way of communicating, behaviour and other traits that make it. — Tiberiusmoon

My understanding of the meaning of the word is that culture is that which is passed down through generations. Animals have instincts and behaviors. They don't pass on what they learn down through generations the way humans do. By this definition animals do not have culture. Of course we (or natural selection) can breed certain behaviors, but that's not the same thing.

I looked up the dictionary definition. "the arts and other manifestations of human intellectual achievement regarded collectively."

The very definition references humans. Animals do not have culture.

I did not claim that physical measurement is exact. — Metaphysician Undercover

Someone using your keyboard, perhaps your cat, wrote

This is an important point for you to recognize. It's not in the real world, (where truth and falsity is determined by correspondence), where the Pythagorean theorem is false, it's tried and tested in the real world, and very true. — Metaphysician Undercover

The Pythagorean theorem in the real world is literally false. It's close but no cigar. It's approximately true, that's the best you can say. But the point here is that you are on record claiming the Pythagorean theorem is "very true." So you are not in a position to deny saying that.

Where I disagreed is with your claim that mathematics has obtained ideal exactness. That is what is factually wrong. Some mathematicians might strive for such perfection, and I would not deny that, but they have not obtained it, for the reasons I described. — Metaphysician Undercover

Idealized mathematics (as opposed to say, numerical methods or engineering math, etc.) is perfectly exact. That's its supreme virtue.

Now it seems to me that the starting point for an interesting discussion is to note that the Pythagorean theorem is literally false in the world, and perfectly exactly true in idealized math; and from there, to meditate on the nature of mathematical abstraction. How we can literally tell a lie about the world, that the theorem is true, and yet that lie is so valuable and comes to represent or model an idealized form or representation of the world.

But if you deny both these premises, one, that the P theorem is false in the world (close though it may be) and perfectly true in idealized math, then there is no conversation to be had. And for what it's worth, your opinions on these two statements are dramatically at odds with the overwhelming majority of informed opinion.

Principally, mathematics has a relationship of dependency on physical world measurements which I described. — Metaphysician Undercover

Of course. "Inspired by." Just as the great work of fiction Moby Dick was inspired by the true story of the Essex, a whaling ship sunk by a whale.

Nonetheless, Moby Dick is a work of fiction. A valuable one, I might add. Fiction is often valuable. Moby Dick teaches us not to follow our obsessions to our doom. That contradicts a point you made earlier that I didn't get a chance to comment on. I believe you said that fiction is always bad, that lies about the world are always bad. Math consists of lies about the world. Nothing in pure math is literally true about the world. Yet fiction, and fictional representations, tell a greater truth by their lies.

THAT is an interesting topic of conversation. Not claiming that the Pythagorean theorem is true in the world and false in idealized math, both claims contrary to fact.

This has ensured that the imprecision of physical world measurements has been accepted into the principles of mathematics. — Metaphysician Undercover

Of course, statisticians have a highly developed theory of measurement error. What of it? Idealized math is inspired by the world.

The lofty goal of idealized exactness has always been, and will continue to be, compromised by the need for principles to practise physical measurement, where idealized exactness is not a requirement. — Metaphysician Undercover

The fact that Moby Dick changed the name of the ship from the Essex to the Pequod, changed the names of the characters, and invented episodes and stories that never really happened, does not detract from the novel in the least. A representation or abstraction stands alone. We do not denigrate the Pythagorean theorem for the "crime" of being exact, when the real-world approximations that inspired it are not. But you so deeply disagree with this point of view that there's little point in continuing. We're just repeating our mutually incompatible premises at this point.

Therefore mathematics will never obtain idealized exactness. — Metaphysician Undercover

It obtains it every day of the week. It obtained idealized exactness in the time of Euclid. Euclid perfectly well understood that his lines and planes and angles were idealized versions of things that did not actually exist in the real world. How you fail to agree to this point of view I don't know. What's important about Euclid is first, the idea of deriving mathematical truths from premises, or axioms; and two, the process of abstraction, meaning that those premises are, strictly speaking, absolute falsehoods about the world. There are no dimensionless points, lines made up of points, and planes made up of lines in the world. Euclid showed how to start with abstract falsehoods (inspired by the world but not literally true about the world); derive logical consequences from them; and thereby obtain insight into the world. Perhaps you should consider that. I can't argue with someone who denies the power of mathematical abstraction.

Look at the role of infinity in modern mathematics — Metaphysician Undercover

I have spent a fair amount of time in my life doing exactly that.

for a clear example of straying from that goal of idealized exactness. — Metaphysician Undercover

The mathematical theory of infinity is a classic example of an abstraction that has nothing at all to do with the real world. And yet, without the mathematical theory of infinity we can't get calculus off the ground, and then there's no physics, no biology, no probability theory, no economics. So THAT is the start of an interesting philosophical conversation. How does such a massive fiction as transfinite set theory turn out to be so darn useful in the physical sciences? Where's Eugene Wigner now that we need him?

But you don't want to have that conversation because you want to utterly reject transfinite set theory simply because it's not literally true about the world. But that's such a boring and trite point of view. Of course it's literally false about the world. The more interesting conversation is to ask how it can nonetheless be so supremely useful in the world. It's the same question as how Euclid's idealized points, lines, and planes can be so useful.

How can lies, in the form of idealized abstractions, lead to truth? That's a good question. Stopping your thought process because the abstractions aren't literally true is not very interesting.


While I've got you here, I wanted to mention that in another thread someone pointed me to Quine's great essay On What There Is (pdf link]. There is a passage that jumped out at me:

If I have been seeming to minimize the degree to which in our philosophical and unphilosophical discourse we involve ourselves in ontological commitments, let me then emphasize that classical mathematics, as the example of primes between 1000 and 1010 clearly illustrates, is up to its neck in commitments to an ontology of abstract entities. Thus it is that the great mediaeval controversy over universals has flared up anew in the modern philosophy of mathematics. — Quine

[My emphasis]

Is this a reference to what you've been trying to talk to me about from time to time? Universals, and how they bear on mathematical abstraction? What does it mean, exactly? After all I frequently point out to you that mathematical ontology posits the existence of certain abstract entities, and this is exactly what you deny. If I understood this point about universals better (or at all, actually) I'd better understand where you're coming from.

Beats me. I was a math prof for years and never had an interest in seeing math in everything. :chin: — jgill

:100:

Reminds me of those people who memorize digits of pi. The only people with zero interest in doing that are math majors.

I don't think everything is math. The most important aspects of life are not quantifiable and not subject to logical or rational analysis.

So they work together: the therapist determines what is needed including medical tests, — tim wood

Uh oh. This is one of those flashpoint conversations that I shouldn't get involved in, because I actually haven't followed it in years.. It's just some stuff I read about 20 years ago.

If you have diabetes, you go in for a blood test and they tell you if you have high blood sugar. Then your doctor prescribes medications to keep your blood sugar under control.

If you have depression, you are told by the psychiatric establishment that you have "low serotonin," and you are given SSRIs. The problem is, there is no blood test for low serotonin. That's contrary to the statement you made about medical tests. The truth is you DON'T get a medical test to determine if an SSRI would help you.

There's an extensive literature on this subject, and like I say my knowledge comes from reading up on this many years ago and not having any ongoing interest. But as far as I know, it's not true that psychiatrists send you in for medical tests before prescribing powerful mind-altering drugs. One could note that all these 20-something boys who shoot up movie theaters and schools are invariably on SSRIs, and that immediately after every such event, politicians talk about guns but never about psych drugs. Another flashpoint conversation.

Are the kids crazy therefore they get put on psych drugs so it's no surprise that kids who do crazy things turn out to have been on psych drugs? Or do the psych drugs make some people crazier than they were before? Far too few inquiring minds want to know. Easier to blame guns, or bad parenting, or trench coats. You remember how schools nationwide banned trench coats after Columbine. No wonder the kids are crazy.

So my original point stands, that it's totally unclear what psychiatrists do. If there IS a medical test showing something wrong with your brain, you need a neurologist. Psychiatrists typically prescribe drugs for conditions that are NOT detectible by medical tests. That's an issue that many anti-psychiatry proponents bring up.

The fact that you believe that mathematics deals with "idealized exactness", is the real problem. Look at the role of things like irrational numbers and infinities in conventional mathematics, these are very clear evidence that the dream of "idealized exactness" for mathematics is just that, a dream, and not reality at all, it's an illusion only. Idealized exactness never has been there, and probably never will be there. — Metaphysician Undercover

@Meta, I'm going to withdraw from this phase of our ongoing conversation. Perhaps we'll pick it up at some time in the future. If you can't agree that real world measurement is necessarily imprecise and that mathematical abstraction deals in idealized exactness, we are not using words the same way and there is no conversation to be had. I don't think you would be able to cite another thinker anywhere ever who would claim that physical measurement is exact. That's just factually wrong.

I remember very clearly a doctor friend of mine telling me that psychiatric illnesses were to be considered as a diagnosis only if organic brain syndrome is/was ruled out as impossible. — TheMadFool

That's right. My understanding of this issue is that on the one hand, if you have something physically wrong with your brain, you need a neurologist; and if you are having emotional difficulties adapting to life, you need a therapist. If you want to run a rat through a maze or convince a population to go to war or smoke cigarettes (think Edward Bernays, who realized that these two were the same problem) you need a psychologist.

In no case do you ever need a psychiatrist, and it's unclear exactly what they do.

Paging Doctor Freud!

My brain neither gains mass nor increases in volume. Ergo, my thought about Aphrodite isn't matter! — TheMadFool

Ahem. When you think of Aphrodite, it's not your brain that gains mass.

Couldn't resist.

This is an important point for you to recognize. It's not in the real world, (where truth and falsity is determined by correspondence), where the Pythagorean theorem is false, it's tried and tested in the real world, and very true. It's only in you imaginary world, of so-called pure abstraction, where the only test for truth is logical consistency, or coherency, that it appears to be false. All this indicates is that your imaginary world is not to be trusted, as it does not give us coherency between even the most simple mathematical principles. On the other hand the Pythagorean theorem alone, can be trusted, because it does give us the right angle. So the quest for logical consistency, or coherency, is not a quest for truth.. — Metaphysician Undercover

Complete misunderstanding of the nature of physical science and the inexactness of all physical measurement. You are living in your own world of delusion.

No, you've got that wrong. The Pythagorean theorem is true in the real world, because it works well and has been proven. Where it is false is in your imaginary world. It works very well for me. I use it regularly. That you think my right angle is a wrong angle is a bit of a problem though. We know induction is not perfect, it just describes what is experienced or practised. (Am I spelling practise wrong?) That the Pythagorean theorem is false in your imaginary world which you call "abstraction", is just more evidence that what you call "abstraction" is not abstraction at all, but fiction. — Metaphysician Undercover

I will get to your earlier longer post when I get a chance, a little busy this week.

That you don't understand that all physical measurement is approximate, and that math deals in idealized exactness that does not correlate or hold true in the real world, is an issue I would have no patience to argue with you. You are simply wrong. Physical measurements are limited by the imprecision of our instruments. This is not up for debate. But I do see a relation between your misunderstanding of this point, and your general failure to comprehend mathematical equality.

How about shapes? Shapes can't exist in isolation. They must be molded from something. — hypericin

Shapes seem more like qualities or attributes. Color, temperature, mass, don't exist in isolation. They are attributes of objects. As is shape. I think shape would fall into the category of an attribute or quality. I'm sure there must be some standard philosophical terminology for this.

Holes are properly thought of as shapes. Their only distinction is concavity. — hypericin

Well a hole is not an attribute of an object, unless you look at it that way ... there's a rock of mass such and so, and color such and so, with a hole in it.

I'm not enough of a philosopher to go down this rabbit hole. I'm in over my head ... in a hole, as it were.

Oh, it comes from Quine's On What There Is, if you haven't read it. I have a hand-wavey understanding of it in the sense that I've read a bit of and about Quine. — Moliere

I gave it a skim, very entertaining and interesting. One thing that jumped out at me right away is that Quine is quite a lively and present writer, not a turgid bore like so many philosophical writers are. He entertains you with his narrative, then when you're not looking he makes his highly insightful points. And he's clear. You can understand what he's saying even as he takes you into the murky depths of ontology. Glad you pointed me to it.

I don't mean to dig into Quine — Moliere

Yes I understand that. It's just that I've heard that particular saying before, and I genuinely don't understand it.

It seems you've changed your stance after your exchange with Wayfarer? — Moliere

No, I think holes exist, and so do shadows. There are things that exist and that only appear along with more substantial things. Holes and shadows being the two that come to mind. Ontologically parasitic, what a great phrase.

I think holes exist though. I haven't had a chance to read the SEP article yet. But there's too much math around the question of identifying and counting holes for me to doubt their existence. But it's a tricky question. Also the question raised by Cool Hand Luke is a good one. Before you dug a hole in the ground, was that hole already there, waiting to be dug out? If not, and you dig it out, you end up with a hole and a pile of dirt. You can't say the pile exists but not the hole. They're sort of like the electron/positron pairs that get spontaneously created in quantum physics. They come into existence in pairs. Piles and holes. You can't say one exists and not the other, can you?

But they have the characteristic of only existing within or as a part of something. You can’t have a hole that exists on its own, whereas you can have an object that exists on its own. So a hole has to be an attribute of something, it can’t have independent existence. — Wayfarer

Hmmm. I will have to think about that, it's a good point. SEP has an article on the subject.

https://plato.stanford.edu/entries/holes/

They even made your point: "Holes are ontologically parasitic: they are always in something else and cannot exist in isolation. (‘There is no such thing as a hole by itself’, Tucholsky 1931.)"

Trying to think of other things that are ontologically parasitic. Shadows, say. Shadows definitely exist but never without a thing to block the light. What kind of existence do shadows have? They're a lot like holes. Here's an article on the philosophy of shadows.

https://www.3ammagazine.com/3am/the-philosophy-of-shadows/

It's a review of a book, Seeing Dark Things: The Philosophy of Shadows.

https://www.goodreads.com/book/show/12858037-seeing-dark-things

What other things are like shadows and holes, things with a tenuous claim on existence, things that are ontologically parasitic?

Fuck you fishfry — Metaphysician Undercover

LOL. Should I read the rest? Second time today someone took one of my little jokes too seriously, I'll practice up on my smileys. :smile: :yikes: :cool: :rofl:

Say, did you know that the Pythagorean theorem is false in the real world? What do you make of that?

ps -- Ok I see you did respond to the rest. By claiming the Pythagorean theorem is literally true. I'll respond later to the entire post. But you know, there are no right angles in the real world. That you think there are is a problem. Must I really walk you through the basics of the philosophy of physical measurement? Sigh. And remember: Only one 'e' in judgment. I trust you won't make that error again. Smiley smiley smiley smiley smiley. Jeez.

To be is to be the value of a variable. — Moliere

I confess I have never understood this in the least. Bound variables are part of symbolic representations, not the things themselves. A cat is the value of a bound variable as in "Exists(x) such that x is a cat," but I find this very unconvincing. The cat is a cat long before there are logicians to invent quantified logic. I just don't understand this kind of thinking. Must be me. A lot of this kind of philosophical discourse just goes right over my head.

Is the question whether holes exist? They most definitely do. Mathematically, if you poke a hole in the x-y plane, then loops around the can no longer be contracted to a point. The hole has changed the topology of the plane. Holes are a huge area of study in math. In algebraic topology they try to find clever ways to count the number of holes in an object. Holes are a thing, not just an absence of a thing.

And of course we have the great scene in Cool Hand Luke.

Boss Paul:
That ditch is Boss Kean's ditch. And I told him that dirt in it's your dirt. What's your dirt doin' in his ditch?

Luke:
I don't know, Boss.

Boss Paul:
You better get in there and get it out, boy.

Holes are things. They have existence.

Hahaha sorry, That is a good joke, just hard to tell over the internet. and I really like this idea the more i think of it as i think it has more and more accurate predictive value. — Ben Ngai

They even study logic that contains inconsistencies.

https://en.wikipedia.org/wiki/Paraconsistent_logic

Solomonoff's solution to the problem of induction — litewave

Thanks for the reference, I'll take a look at that.

The law of gravity is the more general statement, saying all things with mass will fall down. The statement that bowling balls will fall down is more specific. Inductive reasoning is to produce a general statement from empirical observations of particular instances. So, the law of gravity as a general statement, is an inductive conclusion. And, bowling balls may or may not have been observed in producing that inductive law, but the law extends to cover things not observed, due to the nature of inductive reasoning, and the generality of what is produced. This is why inductive reasoning gives us predictive capacity. That mathematics is used to enhance the predictive capacity of inductive reasoning is not relevant to this point. — Metaphysician Undercover

The point is that the process of abstraction necessarily, by its very nature, must omit many important aspects of the thing it's intended to model. You prefer not to engage with this point.

It is important, because induction, by its nature, requires observation of particular instances. And you seem to be arguing that there is a type of abstraction, pure abstraction, which does not require any inductive principles. So it is important that you understand exactly what induction is, and how it brings principles derived from observations of particular instances, into abstract formulae. Do you see that the Pythagorean theorem for example, as something produced from practice, is derived from induction? — Metaphysician Undercover

That's fantastic. The Pythagorean theorem is a beautiful, gorgeous, striking, brilliant, dazzling, elegant, sumptuous, and opulent example of an abstraction that is not based on ANYTHING in the real world and that has NO INDUCTIVE CORRELATE WHATSOEVER. I am thrilled that you brought up such an example that so thoroughly refutes your own point.

The Pythagorean theorem posits and contemplates a purely abstract, hypothetical, mathematical right triangle such that the sum of the squares on the legs is equal to the square on the hypotenuse. No such right triangle has ever, nor will ever, exist in the real world.

@Meta this is such a great example. I wish I had thought of it myself. In soccer they call this an own goal, where you kick the ball into your own net and score a point against your own team.

Oh this is good. Just perfect. You made my day.

To make this clear: The exactitude of the Pythagorean theorem is FALSE for every actual right triangle that's ever existed. It's only in pure abstract mathematical space that it's true. So we go from a fact that is NEVER true in the real world, to one that is ALWAYS true in the abstract mathematical world. This is the complete opposite of induction. It's deduction. It perfectly shows the power of pure abstraction to reveal things about the real world while being based on nothing at all of the real world.

I drop a thousand bowling balls, they all fall down. "Bowling balls fall down." That's induction. I observe a thousand, a million, a gazillion, right triangles, and I note that the sums of the squares on the legs is NEVER equal to the square on the hypotenuse, but only sort of close. From that I DEDUCE -- not "induce," I DEDUCE -- that for a perfect, abstract, Platonic right triangle, the theorem is exact.

Meta you are secretly on my side. I knew it all along! Like a double agent I dispatched into the world long ago and forgot was secretly working on my behalf. I welcome you back to the world of pure, abstract mathematics, in which things can be deductively proven true that are NEVER inductively true in the real world.

It is the inductive conclusion, which allows for the derivation, the prediction, which you refer to. As it is a general statement, it can be applied to things not yet observed. It is not the mathematics which provides the capacity for prediction. mathematics enhances the capacity — Metaphysician Undercover

Your own example falsifies this. I can never confirm the Pythagorean theorem in by observation of the world. I can only prove it deductively and never inductively.

The central point is the difference between the inductive conclusion, which states something general, and the modeling of a "thing", which is a particular instance. At this point, we take the generalization, and apply it to the more specific. It must be determined how well the generalization is suited, or applicable to the situation. This requires a judgement of the thing, according to some criteria. — Metaphysician Undercover

No middle 'e' in judgment. I can't take anyone seriously who can't spell.

I think your description of abstraction as missing things, is a bit off the mark. What abstraction must do is derive what is essential (what is true in all cases of the named type), dismissing what is accidental (what may or may not be true of the thing). Now, if order is essential to being a thing, then we cannot abstract the order out of the thing, to have a thing without order, because it would no longer be a thing. — Metaphysician Undercover

I'm sorry, I can't focus. You so thoroughly demolished your own argument with the Pythagorean example that I can't focus on what you're saying.

But getting back to the larger point: A map is a formalization of certain aspects of reality that necessarily falsifies many other things. Just as a set is a formalization of the idea of a collection, that necessarily leaves out many other things. I have a bag of groceries. I formalize it as a set. The set doesn't have order, the grocery bag does. The set doesn't have milk, eggs, bowling balls, and rutabagas, the grocery bag does.

This is not true in a number of ways. First, good abstractions, inductive conclusions, or generalizations, do not lie because they stipulate what is essential to the named type. They speak the truth because every instance of that named type will have the determined property. — Metaphysician Undercover

I think what you are saying is that an abstraction faithfully represents some aspects of the thing, and leaves out others.

So the essence of your argument is that "inherent order" is so tightly bound to the thing, that it can not be separated by an abstraction. I believe that's the core of our difference. Do I have that right?

Let me say that again, because these posts are getting too long and I believe I've found the essence.

The essence of your argument is that "inherent order" is so tightly bound to the thing, that it can not be separated by an abstraction. I believe that's the core of our difference. Do I have that right?


In other words I could not separate, "Fat bearded guy in a red suit who flies around at Christmas time and climbs down chimneys," from the concept of Santa Claus, because the two notions are so tightly bound that to omit one is to forever de-faithfulize the representation.

Am I now understanding your point?

Second, your proposed "mathematical order" is not an abstraction, inductive conclusion, or generalization. You started with the principle that there is a unity of things with no inherent order. — Metaphysician Undercover

Not necessarily in the world, only in the formal model. Which is no problem.

So you have separated yourself from all abstraction, induction, or generalization, to produce a purely imaginary, and fictitious starting point. — Metaphysician Undercover

For purposes of the conversation, yes, I'll stipulate that. So what? It's how mathematical abstraction works. Just like the Pythagorean theorem does the same thing. There is no right triangle in the world that obeys Pythagoras. Only fake, idealized, imaginary, formal, completely-made-up mathematical right triangles do. Euclid would have been glad to explain this point to you. There are no points, lines, and planes. They're pure mathematical abstractions inspired by, but very unlike, certain things in the real world.

You cannot claim that the imaginary, and purely fictitious starting point, of "no inherent order" is a generalization, or an inductive conclusion, or in any way an abstraction of the physical order. You are removing what is essential to "order", by claiming "no order", therefore you have no justification in claiming that this is an abstraction of physical order. — Metaphysician Undercover

Well now you're just arguing about my semantics. If I don't use the word generalization (which by the way I have not -- please note); nor have I used the phrase inductive conclusion. I have not used those phrases, you are putting words in my mouth. I said mathematical models are formalized abstractions, or formal abstractions. Or formal representations. That's what they are. The need not and generally don't conform to the particulars of the thing they are intended to represent. Just as an idealized right triangle satisfies the Pythagorean theorem, and no actual triangle ever has or ever will. Oh what a great example, I wish I had thought of it. Thank you!

Do you recognize the difference between abstracted and imaginary? Imagination has no stipulation for laws of intelligibility, while abstraction does. — Metaphysician Undercover

Well, abstraction is inspired by things in the real world, and imaginary isn't. But both are instances of formal systems. For example a mathematical right triangle is an abstraction, and chess is imaginary.

OK, now lets proceed to look at your imaginary "mathematical order". — Metaphysician Undercover

As I just defined it, mathematical order is abstract and not imaginary, since it's inspired by the order found in nature.

Do you concede as well, that by removing the necessity of order from your "set", we can no longer look at the set as any type of real thing. — Metaphysician Undercover

My gosh, @Meta, have I ever in all the times we've been conversating EVER referred to sets as real things? They're abstract mathematical objects, hence "real" if by real you mean objects of human thought; as opposed to things in nature like rocks. Of course sets are not "real things." In fact unlike most mathematical objects , sets don't even have a definition. Nobody knows what a set is. A set is anything that satisfies the axioms of some set theory; and there are many distinct axiomatic set theories.

I would never call a set real. But I have never TRIED to call a set real. Why on earth do you think you're challenging me with such a silly question? "No longer" look at a set as real? I never did.

Nor is it a generalization, an inductive conclusion, or an abstraction of physical order. It is purely a product of the imagination, "no order", and as such it has no relationship with any real physical order, no bearing, therefore no modeling purposefulness. It ought to be disposed, dismissed, so that we can start with a new premise, a proper inductive conclusion which describes the necessity of order. — Metaphysician Undercover

There are alternative foundations. I don't see how the choice of foundation is troubling you so much. If you don't like sets, try type theory. I'd say try category theory, but you can do set theory within category theory so that's no escape.

But of course that's not what you're saying. You are objecting to the mathematical concept of set. Well a lot of mathematicians have done the same. On far more sophisticated grounds, which is why it would help you to learn some math if you want to throw rocks at it.

But we conceive of sets as abstractions of collections; and for purposes of getting the formalization off the ground, we conceive of sets as having no order; and then we add the order back in via order theory. I truly don't see why you find this troubling, but I'll accept that you do.

The idea of something with "no inherent order" is not an abstraction, as explained above. It is a product of fantasy, imaginary fiction. — Metaphysician Undercover

You say that like it's a bad thing! Ok, imaginary fiction then. But a useful one! Functional analysis and differential geometry are based on set theory, and quantum physics and general relativity are based on FA and DG, respectively. So you can't deny the utility of set theory, even as you rail against its unreality. On the contrary, the unreality is the whole point of abstraction. But you deny it's abstraction. Ok then, fiction. Ok fine, here's SEP on mathematical fictionalism. There's a philosophical school of thought that completely accepts your premise that math is fiction, nevertheless an interesting and a handy one. That's pretty much the philosophy I'm expressing in my posts to you. Though to be fair, some days I'm a Platonist. Both points of view are useful.

So: Yes math is a fiction. A complete lie. Stuff someone dreamed up one day. What of it? It's still useful. Remember the great essay with the perfect title: "The Unreasonable Effectiveness of Mathematics in the Physical Sciences.' Doesn't that just say it all? Math is so fictional, so clearly NOT based on reality, that it's UNREASONABLE that it's so effective. Yet is is.

So nobody's disagreeing with your point. You need to get beyond your point that math is a fiction, to try to come to terms with why it's so useful.

A map is not an abstraction, it is a representation. I see that we need to distinguish between abstraction, which involves the process of induction, producing generalizations, and as different, the art of applying these generalizations toward making representations, models, or maps. Do you see, and accept the difference between these two? We cannot conflate these because they are fundamentally different. The process of abstraction, induction, seeks what is similar in all sorts of different thing, for the sake of producing generalizations. The art of making models, or maps, involves naming the differences between particulars. These are very distinct activities, one looking at similarities, the other at differences, and for this reason abstraction cannot be described as map making. — Metaphysician Undercover

You know, I see that I am no longer even trying to argue that math is based on reality or represents reality. I could, but then you'll just tangle me up in semantics and fine points. A stronger argument is for me to simply agree with you, completely and wholeheartedly, that math is fiction. And useful. So if you have a problem, it's your problem and not mine, and not math's.

I was referring to the principles of "no inherent order", and "infinity", with the claim that these do not formalize anything. I wasn't talking about maps. — Metaphysician Undercover

Well one is hard-pressed to do physics these days without mathematical infinity, even though the world as far as we know is finite. And I take your point about order, that you think order is so tightly bound to "collections of things" that the two concepts can't be separated by any abstraction. But set theory falsifies that claim, since set theory DOES separate collection from ordered collection.

The map analogy is not very useful, for the reason explained above, it doesn't properly account for the nature of inductive principles, abstraction. Generalizations may be employed in map making, but they are not necessarily created for the purpose of making maps. Now the map maker takes the generalizations for granted, and proceeds from there, but must choose one's principles. In making a map, what do you think is better, to start with a true inductive abstraction like "all things have order", or start with a fictitious imaginary principle like "there is something without order"? Wouldn't the latter be extremely counterproductive to the art of map making, because it assumes something which cannot be mapped? — Metaphysician Undercover

Well set theory isn't map making, of course. and so map makers should start by trying to capture the inherent order of the layout of the streets in a city. But set theorists don't have to do that. So the hell with the map analogy then.

Like I say you have now helped me to clarify my thinking. I have a much stronger position. Math is fiction, and it's useful, so what of it?

So, for the sake of argument, we can make the inductive conclusion, all collections which exist in the world have an inherent order. — Metaphysician Undercover

Yes. I'll stipulate that. And all right triangles in the world violate the Pythagorean theorem. Yet the mathematical version of collection, a set, need not and does not have inherent order; and mathematical versions of right triangles necessarily satisfy the Pythagorean theorem.

This is a valid abstraction, based in empirical observation, and it states that what is essential to, or what is a necessary property of, a collection, is that it has an inherent order. Do you agree then, that if we posit something without inherent order, this cannot be a collection? — Metaphysician Undercover

It can't be a real-world collection, accepting your definition that the molecules in the ocean are "inherently ordered" by virtue of where each and ever one is at any particular moment. Likewise real world right triangles violate Pythagoras. Oh what a great example!

It doesn't have the essential property of a collection, i.e. order; therefore it is not a collection. — Metaphysician Undercover

Well a set isn't actually a collection. A set is an undefined term whose meaning is derived from the way it behaves under a given axiom system. And there is more than one axiom system. So set is a very fuzzy term. Of course sets are inspired by collections, but sets are not collections. Only in high school math are sets collections. In actual math, sets are no longer collections, and it's not clear what sets are. Many mathematicians have expressed the idea that sets are not a coherent idea. It would be great to have a more sophisticated discussion of this point. I'm not even disagreeing with you about this. Sets are very murky.

Each map maker, based on the needs of that map maker's intentions, chooses what to include in the map. Abstraction, inductive reasoning, is very distinct from this, because we are forced by the necessities of the world to make generalizations which are consistent with everything. That's what makes them generalizations — Metaphysician Undercover

I never use the word generalizations. I say abstractions. But if you won't let me do that, then I'll retreat to, "Fiction, and so what?"

Perhaps, but I disagree. It's a matter of opinion I suppose. You desire to put a restriction on the use of "see", such that we cannot be sensing things which we do not apprehend with the mind. I seem to apprehend a wider usage of "see" than you do, allowing that we sense things which are not apprehended. So in my mind, when one scans the horizon with the eyes, one "sees" all sorts of things which are not "forgotten" when the person looks away, because the person never acknowledged them in the first place, so they didn't even register in the memory to be forgotten, yet the person did see them. — Metaphysician Undercover

Was this for me? Oh I see that was for @Luke. LOL.

Well. I hope we can shorten this going forward. I think there are some key points.

* You think that inherent order is so tightly bound with the idea of collection, that the two notions can not be separated by any abstraction. Like Santa Claus and the fat bearded guy in the red suit. That's an interesting point.

* You think math is utter fiction. To which I say, Ok, I'm a mathematical fictionalist myself, and what of it? And Wigner makes the same point. Math is so clearly untrue, that it is unreasonable that it should be so effective. This should be a starting point for your thinking, not an end point. Yes math is fictional. I not only don't argue that point, I have been trying for years to get you to see that. You are the one who wants to reify it.

And that whether or not math is "really" a fiction, which frankly is doubtful, it is nonetheless highly useful to adopt that stance when trying to understand it, so as to take math on its own terms. If you try to figure out whether it's "real" you can drive yourself nuts, because the abstractions get piled on pretty high. So it's better just to take it as fiction and learn the rules. as you do when learning chess.

* So therefore perhaps I should stop wasting my time trying to argue that math is inspired by reality, or that math is an abstraction of reality, and simply concede your point that it's utter fiction, and put the onus on you to deal with that. I could in fact argue the abstraction and inspired-by route, but you bog me down in semantics when I do that and it's tedious.

I think these are the key points here.

Honestly, this is kind of boring and a waste of time. Why not help fix my idea instead of just shooting holes it in and making me do the research to fix it. I thought philosophy and science was collaborative to work together to discover stuff not to just shoot down ideas for fun. — Ben Ngai

I made a logic joke. You are a little off target. Do you need me to explain the joke? You said that a theory would explain a lot if true. I pointed out that by the principle of explosion -- the logical principle that a false antecedent implies anything at all -- a theory that's false explains even more. That's a little logic joke. A false theory explains everything. Then I put a smiley at the end to emphasize the jokitude. Ok so it wasn't the funniest joke ever. I can see that. I'm going to branch into a different universe now in which I never mentioned it at all.

But if my lighthearted and clearly unsuccessful attempt at logic humor was unclear, I do apologize.

So, three logicians walk into a truck stop diner. The waitress walks over and asks, "Do y'all want coffee?" The first logician says, "I don't know." The second logician says, "I don't know." The third logician says, "Yes!"

That's another logic joke in case that was unclear. One that takes a moment's thought.

Okay, i don't need to understand every detail of every theory to realize that this explains a lot if true. — Ben Ngai

Of course by the principle of explosion, a theory that's false explains even more! :starstruck:

I guess physicists have a lot of evidence that points to the stability of the known laws? — litewave

Several decades at least. Of course Ptolemy had evidence too. Isn't this just Hume's problem of induction? Old philosophical conundrum. Like the turkey said on Thanksgiving morning, "The farmer's been good to me every day this year ..."