Yet the Dedekind-Cantor continuum is taught in school along with the fact that a point has zero width. So my objections are bang upto date, as far as I can see. — Devans99

I'm focusing on your claim that mathematicians assume but have not justified the methods of analytic geometry.

No-one has yet pointed out any logic/math error in my OP. — Devans99

I'm not a mathematician, but you're using mathematical language in ways that seems to betray a misunderstanding of fundamental mathematical concepts rather than problems with mathematics. Of course, I could always be mistaken.

If I have something wrong, then someone should set me straight, rather than vague hand waving — Devans99

Would you be satisfied with a proof that one can construct a complete ordered field from the axioms of geometry such that the methods of analytic geometry work out? In other words, would you be satisfied with a construction of such a field from purely geometric premises? My hand-waving is an attempt to communicate these ideas without simultaneously writing a textbook on algebra and geometry. You, on the other hand, refuse to understand basic mathematical concepts.

At least a link to your preferred definition of the continuum would be nice — Devans99

I'm satisfied with the real numbers (up to isomorphism) for the purposes of this discussion.

I directed the OP towards a (highly online) reference that explains how mathematicians disarmed his or her objections over a century ago. The date is relevant only because the OP ignored this reference and continued to insist that the methods of analytic geometry are unfounded. The relevant foundations were provided by mathematicians operating around the turn of the 20th century.

I'm going to have to take some time to consider your response.

I'm saying that your objections are more than a century out of date. In order to understand why, you need to learn a bit of synthetic geometry and abstract algebra. I don't see any way around this.

My view seems pretty unpopular, so i will give it a name: apathetic instrumentalism.

In order to understand the justification for analytic geometry, you need to study synthetic geometry. In particular, Hilbert's work on the foundations of geometry. Your questions were resolved more than a century ago. From the axioms of geometry we can construct models of the real numbers that satisfy our intuitive notion of a coordinate system. There isn't much interesting to see here.

I agree with @fdrake and @SophistiCat that I was "confusing the map with the territory." I will try to separate the core of my argument from your well-deserved critique that I was eliding metaphysical issues by reducing questions about modality to questions about formal systems.

The only reason, I think, that metaphysical questions concerning modality seem largely irrelevant to you (in this context) is that you (seem to) think that "determining the set of acceptable answers" has nothing to do with reasoning metaphysically about modality. Note that you expressed this using the word "acceptable"; itself a modality. Reasoning about the sense of "acceptable" there; what is appropriate for the situation is in part extra-logical; you have to think about the logic from the outside while negotiating its axiomatisation to ensure it works well for the sense of modality in question. — fdrake

I was using "acceptable" as a substitute for "satisfying," which I think eliminates the circularity you pointed out. In general, I think the metaphysical questions are settled (modulo pragmatically irrelevant concerns) once you've determined what you want to know about any particular sentence. In your example ("If I punch the laptop screen, it might break"), you can imagine context determining which answers are satisfying, and therefore which metaphysics are appropriate to adopt. For example, an engineer might be satisfied with a description of the screen's material properties. This might not satisfy the philosopher, but it will satisfy anyone asking whether actual punches will break actual screens.

I guess this turns on the question of what sort of an answer we are looking for: descriptive, explanatory or prescriptive. If descriptive, then reducing the subject to a formal modal logic provides a rough sketch of an answer, but it loses much of the meat in the process of reduction, and the result is only approximate at best, because in reality our modal talk/thinking does not perfectly conform to this system. — SophistiCat

The descriptive ones. If someone asks whether the laptop screen would have broken had they punched it, she would typically not be satisfied with the answer: metaphysically, all counterfactual sentences are vacuously true. This doesn't mean there aren't interesting metaphysical questions about modality (some of which have been brought up in this thread), only that questions about the meaning of modal operators in actual sentences uttered by actual speakers are uninteresting once the question "What are you looking for in an answer?" is settled.

It's important to remember that conservatives aren't interested in meaningful debate. The so-called war on truth is not an epistemic war, but a series of political battles in which one side (conservatives, reactionaries) invents justifications for existing inequalities and launders them through liberal notions of rational discourse. You aren't going to convince conservatives that (for example) the state should enact policies to address racial wealth inequalities, because conservatives are only interested in justifying those inequalities and alleviating their historical guilt.

But this is very backwards, at least in part, if you're using a modal logic to represent metaphysical intuitions, the metaphysical intuitions, the axioms and the theorems all interact; a feeding forward of "acceptable answers" into "metaphysical accounts" makes the appropriate metaphysics for a domain rather arbitrary; or if not arbitrary, we consider the acceptable answers through metaphysical arguments, and at that point we're not just talking about the formal structure of a modal logic either. The extra logical considerations in part determine what logical structure seems appropriate to represent them, so do in part the theorems and axioms of the logic. — fdrake

I think this is a mistake. In order to make sense of a phenomenon - modal talk - you pick a simple formal model that captures some of its structure, and then you try to make sense of your model by studying more of its structure and trying to relate it back to phenomenology. This is what's backwards. You shouldn't lose sight of the phenomenology, and don't expect to find in your model any insight that you didn't front-load there. — SophistiCat

The position I want to defend goes something like this. Consider a counterfactual sentence like "If you had hit the baseball, the window would have broken." Before asking whether this sentence is true, you need to restrict the set of possible answers. For example, here are three possible answers that fail to satisfy most people in most circumstances: (a) the sentence is false because the universe is deterministic; (b) the sentence is true because it describes a logically consistent universe; (c) the sentence is true if modern physics is false. The reason these answers are usually unsatisfying is that most people would consider them irrelevant to determining the truth-value of the sentence in most circumstances in which the sentence arises. For example, I can't imagine any of those responses being used to settle a legal dispute between a neighborhood association and a city government building a baseball field in an adjacent park. Of course, you can always imagine situations where these responses are appropriate. My specific contention is that once the set of acceptable answers is determined, the metaphysical and logical questions are mostly settled and usually irrelevant. The "phenomenology" and "intuitions" are mostly determined by asking someone what they're looking for in an answer.

In my own field (computer science), questions about the truth of sentences often devolve into questions about the provability of sentences in various formal systems. It only makes sense to ask questions about truth and provability once you've determined which systems you're interested in. The choice of system encodes various metaphysical and logical commitments, but is usually driven by practical concerns. For example, if you're interested in provable sentences with computable witnesses, you would avoid adding non-constructive axioms to your theory. It turns out that you can't even ask what it means for a sentence to be true or false without first specifying the range of systems you're interested in.

I hope I'm not missing the point of your objections.

Political correctness and the use of euphemism in science has nothing to do with politics. Political correctness is reviled by both left and right. — NOS4A2

The concept of political correctness was invented as satire by the left, adopted by conservatives lacking the self-awareness to realize they were being mocked, and finally appropriated by reactionaries to justify their victomhood complex. The only people decrying political correctness - i.e., the absolute minimum that you can do, as a human being, to accommodate your fellow citizens - are right-wing ideologues seeking to justify existing systems of inequality. Your bigotry is pretty transparent.

Is there any problem if I formalize "any proposition must be true or false" as ∀p(p∨¬p)?
I know that this can be formalized metalinguistically as something like φ∨¬φ, where φ is any fbf of the object language, an I know that this is not syntatically correct in first order logic, but I want to know if I can set my domain of quantification as the set of all fbfs of the language in question, and, if so, if is there any problem with using logical operators over the variables being quantified.
For example, the existence of the set of all the contradictions (C) would imply that ∀p((p∧¬p)∈C), with p varying over the set of all the well formed formulas. In this case, I use the conjunction and the negation operator in p, which is a wff and also the variable of quantification. — Nicholas Ferreira

A theory that permits self-referential definitions is called an impredicative theory. An example of a theory that permits defining a proposition by quantification over the set of all propositions is the calculus of inductive constructions with an impredicative sort of propositions, the basis of the proof assistant Coq. This theory is sound (and consistent) but lacks a decidable proof theory. My point is that given the right choice of logic, there is no problem with impredicative definitions. You simply have to accept the consequence that your logic might not be decidable. This isn't a huge sacrifice, as evidenced by the enormous number of people who successfully use such logics.

The problem with the traditional terminology is that "one-to-one" is ambiguous between injection and bijection. If you think about the etymology of each word, they are very descriptive: iacere means to throw or cast, and the prefixes (from French/Latin) describe the ways elements of the domain are "thrown" or "cast" into the range.

I think I disagree. The best analysis of modal language we possess is possible worlds semantics. By systematically translating modal talk into talk about possible worlds, questions about counterfactuals can be made precise. The question then becomes: which accessibility relations are germane to our universe? The metaphysics are not elided, but simply shifted to your prescriptions for possible answers. If you are satisfied with considering folk models, then the folk understanding of causality is sufficient to interpret the accessibility relation.

In other words, if you want accessible worlds to be imaginable worlds, or physically reachable worlds, or whatever, you should state that beforehand. It doesn't make sense to ask about the meaning of "might" without also specifying what you consider an acceptable answer. But specifying an acceptable answer determines a metaphysics, and therefore circumscribes the set of accessible worlds.

Critical thinking without context is dangerous. — Banno

I think you're missing something about crackpots. For example, I struggled with the concept of compactness when learning topology. For some time, I thought: surely this isn't the natural complement to discreteness when generalizing finiteness to infinite sets. It wasn't until I learned topology from the perspective of computability theory that I understood the concept. Despite lacking knowledge and thinking critically, I don't think I ever became a crackpot about compactness.

Similarly, I read about the Riemann hypothesis before learning complex analysis. It never occurred to me to question whether the problem was actually within my grasp. I knew that thousands of people much smarter and more informed than myself have worked on the problem, and in order to fully understand let alone approach it I would have years of work ahead of myself. On another note, I avoid talking about the foundations of physics because I'm almost completely ignorant about the subject.

For whatever reason, crackpots focus on specific topics: the axiom of infinity, the Riemann hypothesis, P=NP, the ABC conjecture, and so forth. It's rare to find crackpots discussing the axiom of choice, the mean value theorem, the reality of transcendental numbers, or other actually problematic topics in mathematics (in particular, from the perspective of constructive mathematics). It must be a sociological phenomenon, but I'm not sure how to explain the choice of topics. Surely everyone learned mathematics in high school that are highly suspect and revised in more advanced courses; but those usually aren't the topics they choose to target. Anyways, my two cents.

Thanks. I'll just stick to the simple. — TheMadFool

You can't do that.

Can you have a look at what I said below. — TheMadFool

The subtraction operation on infinite cardinals is not well defined. For example, let $2\mathbb{N}$ denote the set of even numbers and $2\mathbb{N} + 1$ the set of odd numbers. It is easy to see that

• $|\mathbb{N}| = \aleph_0$;
• $|\mathbb{N} - \{0\}| = \aleph_0$;
• $|2\mathbb{N}| = \aleph_0$; and
• $|2\mathbb{N} + 1| = \aleph_0$.

However,

• $|\mathbb{N} - (\mathbb{N} - \{0\})| = |\{0\}| = 1$; and
• $|\mathbb{N} - 2\mathbb{N}| = |2\mathbb{N} + 1| = \aleph_0$

Thus, we cannot say that $\aleph_0 - \aleph_0 = 1$ or $\aleph_0 - \aleph_0 = \aleph_0$. There are some technical conditions under which you can define a subtraction operation, but they are beyond the scope of this question.

As with the teaching of infinity, something which is just an assumption is taught to us as absolute knowledge. I feel our maths teachers are letting us down — Devans99

The Cantor-Dedekind axiom is not an axiom in the usual sense. You can construct a complete ordered field over the Euclidean plane from the axioms of synthetic geometry. Since every complete ordered field is essentially the real numbers, this justifies the methods of analytic geometry. This justification required the development of axiomatic methods for geometry and algebra. A classic and highly readable reference is Hartshorne's Geometry: Euclid and Beyond (Chapters 2-3).

There are resurgent white supremacist movements across North American and Europe; white supremacists hold positions of power and influence in governments, police forces, militaries and corporations across those continents; white supremacists have systematically terrorized people of color and other minority groups for centuries. It seems reasonable, therefore, to find a substitute for the word "supremacy" provided that it distresses some non-trivial group of people victimized by white supremacy. This is common courtesy, not a manifestation of some pretended leftist assault on science. The real danger to scientific practice comes from the right (climate denialism, creationism, race science, etc.).

By definition, $\aleph_0$ is the cardinality of the natural numbers. Your argument does not establish that the natural numbers have finite cardinality. I thought that clarifying what people mean when they use the series notation might disabuse you of the notion that there must be some "transition" from "finite numbers" to "infinite numbers" happening.

You've described a potential infinity, but not an actual infinity. To understand an actual infinity we need to understand the actual existence of the elements represented by mathematical language. — Metaphysician Undercover

When you say "the set of natural numbers," you mean "the set of all objects that can be generated from zero and the successor function and which satisfy the Peano axioms" (or something similar). In other words, I can use the language of sets to talk - all at once - about every object satisfying these conditions. If that doesn't satisfy your condition that someone "understand the actual existence of the elements represented by mathematical language," then perhaps we simply have different conceptions of the purpose of mathematical language (e.g., the language of sets).

You are misinterpreting my response. I was responding to your argument by clarifying what people mean when they say things like "the series equals infinity." This is simply a shorthand for "the series diverges" or "the limit of the partial sums does not converge to a real (complex) number." It doesn't follows that cardinal and ordinal arithmetic is "false" (I'm not even sure what that means) or that space and time cannot be unbounded.

I think fishfry said something to the effect that bijection has precedence of injection. Why? — TheMadFool

Hopefully this explains my comment. If $f$ is a function from a set $A$ to a set $B$, then $f$ is

• injective iff $f(x) = f(y)$ implies $x = y$ for all $x$ and $y$ in $A$;
• surjective iff for all $y$ in $B$ there exists an $x$ in $A$ such that $f(x) = y$; and
• bijective iff for all $y$ in $B$ there exists a unique $x$ in $A$ such that $f(x) = y$.

Intuitively, an injection maps unique elements of its domain to unique elements of its range; a surjection has every element of its range mapped to by some element of its domain; and a bijection has every element of its range mapped to by a unique element of its domain. We have the following theorems:

• $f$ is bijective iff it is injective and surjective.
• $f$ is bijective iff it has an inverse.

You should convince yourself of the following facts. If $f : A \rightarrow B$ is

• injective, then there may be some elements of $B$ that are not mapped to by any element of $A$;
• surjective, then there may be some elements of $A$ that map to the same element of $B$; and
• bijective, then there is a unique pairing between elements of $A$ and elements of $B$

These observations justify the following definition of an ordering relation on the cardinal numbers:

• $|A| \leq |B|$ iff there is an injection from $A$ to $B$; and
• $|A| = |B|$ iff there is a bijection from $A$ to $B$

From the above discussion, it should be clear that $|A| \lt |B|$ iff there is an injection from $A$ to $B$ but no bijection. This means that every injection from $A$ to $B$ must fail to be surjective (i.e., for every injection there exist elements of $B$ that the injection misses, although those elements will be different in different cases).

In classical mathematics, we can show that the ordering relation is total. By "classical," I mean mathematics with the axiom of choice. If we assume a constructive framework where the axiom of choice (law of excluded middle, double negation elimination, etc.) fails, the discussion becomes more complicated. IIRC, the "bijection between a set and a proper subset of itself" definition of infinity fails in a constructive setting, you cannot show that the ordering relation is total, an injective function does not imply the existence of a surjection from its image to its domain, and so forth.

I think Enderton's Elements of Set Theory is a good introduction to these topics.

I think step [6] above is no doubt questionable, but it brings out the point: how exactly does a finite number ever become infinite? - We have no basic arithmetical operators to convert finite numbers into infinite numbers. To focus on this aspect, here is a similar argument that more graphically brings out the discontinuity between natural numbers and infinity: — Devans99

The correct terminology is that the series $\sum_{i = 0}^\infty 1$ diverges. In other words, the limit of the partial sums $\sum_{i = 0}^n 1$ does not converge to a number as $n$ tends to infinity. The notation $\sum_{i = 0}^\infty 1 = \infty$ does not mean that the series equals some number called 'infinity' and denoted '$\infty$'.

Now the issue, which we discussed already in the thread, is whether or not a written numeral necessarily represents an object. In actual usage, the numeral might be used to represent an object, or it might not. If it doesn't represent an object, then any supposed count is not a valid count.

Your example seems to create ambiguity between the symbol, and the thing represented by the symbol. So you would have to clarify whether there is actually existing numbers, existing as objects to be counted, otherwise the claim of "an infinite set of objects" is false. As proof, it doesn't suffice to say that it is possible that a numeral represents an object And actual usage of symbols demonstrates that it is possible that the symbol represents an object, but also possible that it does not. To present the symbol as if you are using it to represent an object, when you really are not, is deception. — Metaphysician Undercover

I don't understand why skepticism about the meaning of mathematical language is relevant to the discussion about potential and actual infinities. The notion of potential infinity is the notion of a process that can be repeated indefinitely. For example, one can always consider successively larger values of the Fibonacci sequence. The language of sets (types, classes, etc.) provides one way of talking about the elements generated by such processes. The notation used is entirely irrelevant.

2. A set G has a cardinality greater than a set H if and only if the there's a bijection between set H and a proper subset of G — TheMadFool

This is false. For example, $\mathbb{N}^+$ is a proper subset of $\mathbb{N}$ and the function $f(n) = n - 1$ is a bijection from $\mathbb{N}^+$ to $\mathbb{N}$, but the cardinality of both sets is $\aleph_0$. However, it is classically true that $A \lt B$ iff there is an injection from $A$ to $B$ but no surjection.

I like this, but skills is perhaps better than beliefs, in that 'beliefs' casts the whole thing as more explicit than I think it is. Have you looked into Dreyfus's Being-in-the-world? The 'form of life' is something like a set of norms that aren't explicit and can't plausibly be enumerated. — softwhere

I haven't read Dreyfus, but I'm familiar with Heidegger. In response to your question, I would argue that for Davidson beliefs are behavioral dispositions, as are skills. For a subject to believe that $p$ is simply to be disposed to say $p$ in response to the appropriate simuli. You can't differentiate someone who believes that $p$ from someome who is simply disposed to say that $p$ is true in the appropriate circumstances. In your terminology, beliefs are skills.

EDIT: I said for Davidson, but I'm really reading him through Quine.

Later on he says: "...belief is in its nature veridical." — ZzzoneiroCosm

Davidson is arguing that members of actual linguistic communities have mostly true beliefs about the world. It is relevant to his argument that disagreement about specific facts can only occur against a background of shared true beliefs. It is useful to compare this discussion to Wittgenstein's discussion of hinge propositions in On Certainty, although I suspect there would be disagreement between Davidson and Wittgenstein on certain technical points.

In other words, linguistic communities tend to converge on standard labels for various kinds of stimuli, standard ways of talking about those stimuli, and so forth. It follows that unless someone simply doesn't understand the language, that they will use those labels and ways of talking to communicate information about their environment. Because of this, their responsive dispositions (beliefs) will be mostly similar and mostly accurate, simply because they were developed in response to the kinds of things that elicit those responses. This explanation is derived from broadly Quinean arguments about the nature of belief rather than details about the principle of charity, anomalous monism, or whatever specific philosophical theses Davidson has proposed.

I'm confused by the distinction actual vs potential infinity?

From wikipedia I get:

Potential infinity is a never ending process - adding 1 to successive numbers. Each addition yields a finite quantity but the process never ends.

Actual infinity, if I got it right, consists of considering the set of natural numbers as an entity in itself. In other words 1, 2, 3,.. is a potential infinity but {1,2, 3,...} is an actual infinity.

In symbolic terms it seems the difference between them is just the presence/absence of the curly braces, } and {.

Can someone explain this to me? Thanks. — TheMadFool

The principle of induction is intimately related to the recursion theorem. The principle of induction states that if $\phi$ is a predicate such that $\phi(0)$ and $\phi(n) \rightarrow \phi(n + 1)$ for all $n \geq 0$, then $\phi(n)$ is true for all natural numbers $n$. A proof of the induction principle is a recursive function that transforms a proof of $\phi(0)$ and a proof of $(\forall n \geq 0)(\phi(n) \rightarrow \phi(n + 1))$ into a proof $(\forall n)\phi(n)$ for any predicate $\phi$.

The induction principle is interesting because it allows us to conclude that $\phi(n)$ is true for every natural number $n$ without checking that $\phi(n)$ is true for every natural number $n$. In other words, it allows us to conclude that every member of an infinite set of objects possesses some property by performing a finite amount of work. There are analogous induction principles for other infinite sets. For example, some proof assistants automatically generate induction principles for arbitrary datatypes satisfying certain technical conditions.

In other words, there is a systematic relationship between certain infinite sets and inductive proofs (recursive functions), the latter of which are "potential infinities" in your terminology. In other words, induction principles (recursive functions) systematically translate between statements about "potential infinities" and statements about the elements of certain infinite sets. This has nothing to do with curly braces or other arbitrary syntactic phenomena unless you are committed to some kind of hardline formalism about mathematics.

Here is another example: consider the "potential infinity" defined by the Fibonacci sequence. You can generate every Fibonacci number using a recursive function defining the sequence. In other words, the recursive function defines the first, second, third, and so forth, Fibonacci number. However, you can always consider the collection of elements generated in this way by saying: "suppose that $n$ is a number in the Fibonacci sequence." What you are talking about, in the latter case, is an infinite set of objects - there is no limit to the number of objects that satisfy this condition, although there are restrictions on the kinds of objects that satisfy the condition.

The simple answer is that "Mary might buy ice cream" is true iff there is an accessible world in which Mary buys ice cream. The heart of your question is: what is meant by an accessible world? The simplest answer is: a possible world $w$ is accessible from our world iff $w$ can be imagined without contradiction. A more restrictive answer is: $w$ is accessible from our world iff $w$ can be obtained from our world through physical processes. A more metaphysically adventurous answer is: $w$ is accessible from our world iff $w$ exists. The last answer probably commits you to strong forms of modal realism. In general, I think you need to make your question more precise in order to elicit interesting discussion.

However, let's do something different. We take the same sets N and E. We know that N has the even numbers. So we pair the members of E with the even numbers in N. We can do that perfectly and with each member of E in bijection with the even number members of N. What now of the odd numbers in N? They have no matching counterpart in E. — TheMadFool

The problem with your argument is that the described function is not bijective. Instead, you have constructed an injection from the set of even numbers to the set of natural numbers. From this, you are only permitted to conclude that the set of even numbers has at least as many elements as the set of natural numbers. If you combine this with an injection from the set of natural numbers to the set of even numbers, then you can conclude that a bijection exists. Because these sets are countable, this bijection is constructible (and is given by the standard map from the natural numbers to the even numbers).

I have never seen necessitation presented as an axiom. Rather, it is usually presented as a deduction rule. It states that if $\phi$ is a theorem of a modal logic $\mathbb{K}$, then $\square\phi$ is a theorem of $\mathbb{K}$. In other words, every theorem of the logic is necessary. In epistemic logics, the necessitation rule states that every agent knows every proposition derivable from no assumptions (i.e., every agent knows every tautology). This captures the idea that epistemic agents are, or ought to be, logically consistent.

This is a mostly geometric argument and it goes like this. — Umonsarmon

The problem with your algorithm is that the binary representation of a natural number is finite. Therefore, your algorithm picks out only those numbers that can be expressed through finite bisections of an interval. It turns out that the unit interval is isomorphic to the set of infinite binary sequences. The diagonalization argument establishes that the set of finite binary sequences does not surject the set of infinite binary sequences. It therefore deserves the name "uncountable." However, because there is an injection from the set of finite binary sequences to the set of infinite binary sequences that does not surject the set of infinite binary sequences, this "uncountable" set is strictly larger than the "countable set," despite the obvious fact that both sets are infinite.