I say the US education system does a massive disservice to the field of mathematics due to the fact that it divorces the philosophy of mathematics away from the applied version. — Vaskane

the philosophy of mathematics in the 20th century was characterized by a predominant interest in formal logic, set theory (both naive set theory and axiomatic set theory), and foundational issues.

In fact, this was attempted in the New Math of the 1957- 1970s. It was a disaster. For a variety of reasons. I know, I was there in the classroom.

For me, as a kid, New Math was wonderful. It opened my imagination to different ways of looking at mathematics, not just learning by rote how to do long division and stuff like that. For example, the idea of numbers in binary, modular arithmetic, intersections and unions, truth tables. I think maybe the idea behind it was to get children primed for the upcoming age of computers, such as binary numbers. It blew my mind, I savored it and it served me well.

, . Perhaps the emphasis on imagination in some maths teaching is what leads to those here who insist on the truth of their mathematical intuitions despite demonstrations that they are mistaken. Is @Metaphysician Undercover a product of the New Maths? :wink:

The problem for curricula is that there is of course no one way to teach, each individual having different needs and backgrounds. What is required is trust in the teacher's ability to recognise and adapt their teaching to the student. But that's contrary to the very notion of a curriculum.

What is required is trust in the teacher's ability to recognise and adapt their teaching to the student. But that's contrary to the very notion of a curriculum. — Banno

That seems to me to be a trenchant observation.

except it can be used as an adjectiv — Vaskane

:brow:

What is one sentence where "infinity" is used as an adjective?

What is one sentence where "infinity" is used as an adjective? — Lionino

My cousin spent $85,000 on an infinity pool because he thinks that if he swims in it he will live forever.

My cousin has an infinity pool because he thinks that if he swims in it he will live forever. — TonesInDeepFreeze

That is typically called an open compound noun in English. If we are doing a semantic analysis, infinity pool, the 'infinity' qualifies 'pool' implying that it looks endless, so 'infinity' would stay a noun and be in the similative case — English does not have grammatical cases morphologically, but semantically that is what it would be. English does not have morphological rules for adjectives (or for any word class I think), so the '-y' ending does not stop 'infinity' from one day becoming an adjective, but it is just not used as such today.

So he spent $85,000 for just a fancy noun. I told him it was not a wise purchase.

Of course, but I'm saying that in context of sets in mathematics, 'infinity' as a noun invites misunderstanding, especially as it suggests there is an object named 'infinity' that has different sizes. — TonesInDeepFreeze

In the Merriam Webster dictionary infinity is classed as a noun, and within mathematics is the infinity symbol ∞. But as you say, this is problematic as it suggests that infinity is an object, such as a mountain or a table, which can be thought about. But in one sense this is impossible, as it impossible for a finite mind to know something infinite, where infinity is an unknowable Kantian "ding an sich" ("thing in itslef").

So what does the word "infinity" refer to, if not a noun inferring an object?

As the Wikipedia article on Infinity writes: Infinity is a mathematical concept, and infinite mathematical objects can be studied, manipulated, and used just like any other mathematical object.

As Literature as a noun refers to the study of books, perhaps Infinity as a noun refers to the study of infinite sets. Both Literature and Infinity are nouns, but refer to a process, not the intended conclusion of the process. This makes sense, as processes are comprehensible to finite minds. A finite mind can comprehend the process of adding to an existing set, even if not able to comprehend the eventual conclusion of continually adding to an existing set .

IE, "infinity" is a noun and refers to a process rather than any conclusion of that process.

One is free to say that we don't need utter objectivity, but then we may say, "Fair enough. So your desideratum is different from those using the axiomatic method." — TonesInDeepFreeze

My statements were from my reasoning. But what you claim to be objectivity is from the textbooks. Please bear in mind, the textbooks are also written by someone who have been reasoning on the subject. It is not the bible, to which you have to take every words and sentences as the objectivity that everyone on the earth must follow. That sounds religious.

Mathematics is a narrow scoped subject which borrowed most of its concepts from Philosophy and modified to suit their abstraction to justify their theorems. Hence we find lots of confusions in math and also the math students. Philosophy can clarify some of its modified concepts for the real meaning of them, so they can understand the subject better.

Infinity pools can indeed be awesome: :starstruck:

As a "set" is an object it can have a size, and therefore there can be different sizes of sets.

However, as the qualifier "that can be added to" is not an aspect of the size of the set, whilst the expression "different sizes of sets" is grammatical, the expression "different sizes of infinite sets" is ungrammatical. — RussellA

The whole confusion resulted from the wrong premise that infinite numbers do exist. No they don't exist at all. So it is an illusion. From the illusive premises you can draw any conclusions which are also illusive.

Infinity in math has been improvised to explain and describe continuous motion hence the Limit and Integral symbols in Calculus. But they have taken the concept further to apply into the set and number theories. Yes depending on what you accept, you can say the infinite Sets can have different sizes etc. It is OK to keep on saying that in math forums, and it sounds correct because that is what the textbook says.

But when it comes to under the Philosophical analysis, one cannot fail to notice the whole picture was based on the fabricated concepts, which are not very useful or practical in the real world.

The whole confusion resulted from the wrong premise that infinite numbers do exist. No they don't exist at all. So it is an illusion. — Corvus

:up: :up: :up:

Some here might like finitism or ultrafinitism. Wikipedia has a page, and there's a more technical intro here: nlab. The following is about an extreme ultrafinitist.

I have seen some ultrafinitists go so far as to challenge the existence of $2^{100}$ as a natural number, in the sense of there being a series of “points” of that length. There is the obvious “draw the line” objection, asking where in $2^1, 2^2, 2^3 \dots , 2^{100}$ do we stop having “Platonistic reality”? Here this … is totally innocent, in that it can be easily be replaced by 100 items (names) separated by commas. I raised just this objection with the (extreme) ultrafinitist Yessenin-Volpin during a lecture of his. He asked me to be more specific. I then proceeded to start with $2^1$ and asked him whether this is “real” or something to that effect. He virtually immediately said yes. Then I asked about $2^2$, and he again said yes, but with a perceptible delay. Then $2^3$, and yes, but with more delay. This continued for a couple of more times, till it was obvious how he was handling this objection. Sure, he was prepared to always answer yes, but he was going to take $2^{100}$ times as long to answer yes to $2^{100}$ then he would to answering $2^1$. There is no way that I could get very far with this. — Harvey Friedman, Philosophical Problems in Logic

I like the one on the top, but not the one on the bottom. I enjoy big cities more than the next guy, but something about pools with sunbathing beds (and worse! palm trees) ontop of a building on a big city rubs me the wrong way.

I have seen some ultrafinitists go so far as to challenge the existence of 2 100 2 100 as a natural number, in the sense of there being a series of “points” of that length — Harvey Friedman, Philosophical Problems in Logic

That reminds me of intuitionists or at the very least of psychologists in the ontology of mathematics, where the number 2^100 does not exist until it is thought up.

Is Metaphysician Undercover a product of the New Maths? :wink: — Banno

I do not think I was ever subjected to new math. I simply learned at a very young age not to follow rules without a reason for doing so. I was not interested in the things which mathematics was useful for, so it was dismissed from my curriculum, as soon as possible, until the need was developed. So my education in mathematics was done in an 'as required' way, rather than a force-feeding of conventional 'fact' to memorize, like history.

Infinity pools can indeed be awesome — ssu

You get the same effect when you take a boat on a reservoir, up toward the dam, the higher the dam the better. It's like empirical proof that the earth is flat, and you're at the edge of the world.

That reminds me of intuitionists or at the very least of psychologists in the ontology of mathematics, where the number 2^100 does not exist until it is thought up. — Lionino

But then again, the number 2 does not exist until it is thought up.

I have seen some ultrafinitists go so far as to challenge the existence of 2 100 2 100 as a natural number, in the sense of there being a series of “points” of that length
— Harvey Friedman, Philosophical Problems in Logic

That reminds me of intuitionists or at the very least of psychologists in the ontology of mathematics, where the number 2^100 does not exist until it is thought up. — Lionino

This reminds me of the axiomatic systems that perhaps some animals (or people) have: nothing, 1, 2,3,4, many. When you think of it, it's quite useful for up to a point.

You get the same effect when you take a boat on a reservoir, up toward the dam, the higher the dam the better. It's like empirical proof that the earth is flat, and you're at the edge of the world. — Metaphysician Undercover

Yes, I've always pondered how few people go to a port or to the seashore and simply look at how large ships simply "sink" into the horizon far earlier than they become tiny specs. But I guess flat Earthers just have this habit of going with the crazy and being against the tyrannical science & math we "sheeple" so blindly accept and follow. It makes them special.



And because the math is extremely hard:

And note it's called a theorem.

But as you say, this is problematic as it suggests that infinity is an object, such as a mountain or a table, which can be thought about. — RussellA

I don't say that.

I say that 'infinity', applied to set theory, is not advisable, because in set theory there is no object called 'infinity', especially one that has different cardinalities. It's not a matter of can be thought about, but rather that there are many infinite sets, not just one called 'infinity'.

within mathematics is the infinity symbol ∞ — RussellA

The lemniscate is usually used to indicate a point of infinity on a number line, which is very different from the context of the cardinalities of infinite sets. Such a point of infinity is some designated (or sometimes, less formally, unspecified) object along with a set, such as the set of real numbers, and an ordering is stipulated. If the treatment is fully set theoretical, then the object itself can be infinite or not.

So what does the word "infinity" refer to, if not a noun inferring an object? — RussellA

I am not saying that one should not use 'infinity' as a noun. It is a noun. And people can use it for many things. But it is an invitation to confusion to use 'infinity' regarding set theory or mathematics in a context such as discussing infinite cardinalities. Set theory does not define an object named 'infinity' in this context. Rather, it defines a property 'is infinite'. Keeping that distinction in mind goes a long way to avoiding confusions.

As the Wikipedia article on Infinity writes: Infinity is a mathematical concept, and infinite mathematical objects can be studied, manipulated, and used just like any other mathematical object. — RussellA

(1) There are better sources than Wikipedia.

(2) The quote does not say that mathematics refers to some set that is named 'infinity'.

(3) The quote then defers to 'infinite', the adjective, which is correct.

(4) The article is almost all about 'infinity' not applied to infinite sets and cardinalities. And the small part of the article that is concerned with infinite sets and cardinalites correctly, when talking about sets, uses 'is infinite', the adjective for the property of being infinite, not 'infinity' to name a set.

Infinity in math has been improvised to explain and describe continuous motion — Corvus

Also, more simply, there are infinite sets of numbers, such as the set of naturals, the set of rationals, and the set of reals. With ordinary classical logic, for even just simple first order PA to have a set over which the quantifier ranges requires an infinite set.

Sets can have different sizes etc. It is OK to keep on saying that in math forums, and it sounds correct because that is what the textbook says. — Corvus

Not just because it's what a book says. Rather, textbooks provide proofs of theorems from axioms (including definitional axioms) with inference rules. One doesn't have to accept those axioms and inference rules, but if one is criticizing set theory then it is irresponsible to not recognize that the axioms and inference rules do provide formal proofs of the theorems. Moreover, intellectual responsibility requires not misrepresenting the mathematics as if the mathematics says that the theorems claim simpliciter such things as that there are infinite sets of physical objects or even that there are infinite sets in certain other metaphysical senses of 'infinite'.

the whole picture was based on the fabricated concepts, which are not very useful or practical in the real world. — Corvus

Fabricated in the sense of being abstract. And it is patently false that classical infinitistic mathematics is not useful or practical. Reliance on even just ordinary calculus is vast in the science and technology we all depend on.

I do not think I was ever subjected to new math. — Metaphysician Undercover

Ok. So we still have no explanation of how you came to misapprehend "=".

But what you claim to be objectivity is from the textbooks. — Corvus

What I said was that it is objective to mechanically check that a purported formal proof is indeed a proof from the stated axioms and rules of inference. If there is anything more objective than verification of application of an algorithm, then I'd like to know what it is.

It is not the bible, to which you have to take every words and sentences as the objectivity that everyone on the earth must follow. — Corvus

Of course. And I have many times explicitly said that no one is obligated to accept, like, or work with any given set of axioms and inference rules. But if the axioms and inference rules are recursive, no matter what else they are, then it is objective to check whether a given sequence purported to be a proof sequence is indeed a proof sequence per the cited axioms and rules. If you give me formal (recursive) axioms and rules of your own, and a proof sequence with them, then no matter whether I like your axioms or rules, I would confirm that your proof is indeed a proof from those axioms and rules.

we find lots of confusions in math and also the math students — Corvus

There are areas of great puzzlement and disagreement in the philosophy of mathematics. But I don't know what specific confusions you refer to, specifically in formalized classical mathematics.

the expression "different sizes of infinite sets" is ungrammatical. — RussellA

I don't think so. And it's clear to me. There are infinite sets that have sizes different from one another. More formally:

There exist x and y such that x is infinite, and y is infinite, and the size of x is not the size of y.

The whole confusion resulted from the wrong premise that infinite numbers do exist. — Corvus

What is the "whole confusion"? Yes, there are people who don't know about set theory and are confused about it so that they make false and/or confused claims about it. But the axioms of set theory don't engender a confusion. They engender philosophical discussion and debate, but there is no confusion as to what is or is not proven in set theory. Whether any given axiom is wrong or not is a fair question, but it doesn't justify people who don't know anything about axiomatic set theory thereby spreading disinformation and their own confusions about it.

I do not think I was ever subjected to new math. — Metaphysician Undercover

Virtually any student is subjected to certain instruction whether they like it or not. It would be fair to say that New Math is not good only if one at least knows what it is.

my education in mathematics — Metaphysician Undercover

You have virtually no education or self-education in the mathematics you so obdurately opinionate about.

What's worse, a population of palm trees in a city, or a city in a population of palm trees?

you hold a boat load of mathematical knowledge — Vaskane

My knowledge in mathematics is quite meager compared with people more dedicated to the study.

the US education system does a massive disservice to the field of mathematics due to the fact that it divorces the philosophy of mathematics away from the applied version. — Vaskane

That might be true. Also, the fact that formal logic is not conveyed so that students could see how the mathematics is derived logically rather than simply decreed.

One can always learn vastly more about logic, mathematics, philosophy and the philosophy of mathematics, but my first interest in logic (which led to mathematics) came from my interest in philosophy, and as I learned logic and mathematics, I was learning about the philosophy of mathematics right alongside.

Ok. So we still have no explanation of how you came to misapprehend "=". — Banno

Did you not understand the example I gave you in the other thread? I suggest you go back and read that post you made for me when you fed that example to Chat GPT. It totally agreed with me. It said, in much arithmetic and mathematics "=" signifies equality, not identity. Chat GPT does not lie you know. The simple fact, as my example shows, an "equation" would be completely useless if the left side signified the very same thing as the right side.