• Anscombe's "Modern Moral Philosophy"
    ↪Banno


    Learning how to cultivate it is not. "Never do for a patient what they can do for themselves" was discovered.
  • Anscombe's "Modern Moral Philosophy"
    ↪Banno


    Dog barks at kid and approaches, dog is all waggy and friendly. Kid is scared, kid's never been near a dog before. "Don't worry, they're just excited to see you, be calm and slowly put your hand out." Kid does it, has dog pats. "Well done, you were courageous". An important learning experience, why?

    "Some questions can't be answered and that will have to do."

    Not good - nowhere near bedrock here.
  • Anscombe's "Modern Moral Philosophy"
    Well, the point of virtue ethics was to avoid rules, so... — Banno

    (I agree) What do you think the objects of it are, then? EG: People and properties?
  • Anscombe's "Modern Moral Philosophy"
    Virtue ethics strikes me as providing a much richer environment for teaching than, say, deontology. Compare punishing someone for breaking the rules with cultivating kindness, courage and resilience. — Banno

    This takes us outside the text.

    A common criticism (IIRC) is that virtue ethics provides no rules that engender moral behaviour. I quite like this; as if being moral or living well wasn't (1) very contextual and (2) leant on other sources of knowledge (3) a perpetual process of (interpersonal/skill set) growth (and mitigation of decay). There's no "special domain of problems" associated with virtue ethics; its problems are just those life happens to throw up.

    Edit: well, that's intended idea, there are of course philosophical problems, virtue in the face of randomness (moral luck) etc.

    This immerses back in the text:

    The analysis of "relative brute facts", the appeal to both social institutions and natural properties when considering what it means to do an injustice ("bilking") and a latent "naturalisation of the connection of need to ought" (scare quotes because it's not in the text) speaks to the contextual nature of ethics in (my interpretation of) Anscombe's vision of it. Allying virtue ethics with conceptual analysis rather than vouchsafing the origin of prescriptive morality (warrant and nature of ought claims) makes it more of a tool or an elaboration on life. In one breath it's empty of (allegedly more standard) philosophical problems, but that's because its domain is as broad as human action and leverages all that goes into it.
  • Anscombe's "Modern Moral Philosophy"
    Why invoke eudaemonia? — Banno

    To sell self help books.

    To learn to cultivate yourself and others.

    To leverage that understanding for political criticism and intervention.
  • Mathematicist Genesis
    (If anyone's actually reading along, questions are fine too)
  • Anscombe's "Modern Moral Philosophy"
    There are a few bits in the paper that allude to the role moral psychology should play in a new conception of moral philosophy.

    But in the case of a plant, let us say, the inference from "is" to "needs" is certainly not in the least dubious. It is interesting and worth examining; but not at all fishy. Its interest is similar to the interest of the relation between brute and less brute facts: these relations have been very little considered. And while you can contrast "what it needs" with "what it's got"-like contrasting de facto and deiure-that does not make its needing this environment less of a "truth." "

    (whatever the relative brute-ness construction does) seems to be proposed as a logical mechanism for grounding (whatever the new conception Anscombe has about moral philosophy) in (some conception of the psychology of morals and human nature).

    Certainly in the case of what the plant needs, the thought of a need will only affect action if you want the plant to flourish. Here, then, there is no necessary connection between what you can judge the plant "needs" and what you want. But there is some sort of necessary connection between what you think you need, and what you want. The connection is a complicated one; it is possible not to want something that you judge you need. But, e.g., it is not possible never to want anything that you judge you need. This, however, is not a fact about the meaning of the word "to need," but about the phenomenon of wanting. Hume's reasoning, we might say, in effect, leads one to think it must be about the word "to need," or "to be good for."

    Allegorically, plants need water and sunlight because of what plants are. If moral philosophy was done by plants, maybe a Hume plant would have written that: "Whenever I have occasioned to find a treatise regarding our affairs, I have observed a transition from the usual copulations of propositions like "is a water requiring organism" and "does not survive without water" to "ought to absorb water " and "ought not station themselves in an arid landscape" which express a new affirmation of a different sort. Wherefrom this new type of affirmations are deduced from their precedents despite this difference in character I cannot conceive".

    Such a plant could be alleged to engender a discourse about plant morality that concerned the operation of the word "ought" severed from the context of how plants behave and develop; more concerned with an abstract rule of application of these words to statements of fact regarding plant behaviour and development. Anscombe would want the plants to talk about how they need water because of their metabolic requirements than about a moral logic operating on statements of facts concerning plants.

    So part of the appeal to moral psychology is to explain why such a move appears to make sense; and Anscombe finds this in divine law and legalist accounts of moral obligation, in historical context. But, Anscombe suggests, such motivating contexts have long since lost their ability to furnish our understanding of morality, and that moral philosophers simply have not kept up with the times. More specifically, legalistic/contractual and divine law conceptions of morality stymie the formation of any conception of morality which is tethered to real life rather than dead metaphors; and moral philosophy in the criticised senses has replaced these dead metaphors with philosophical principles that function to fill void left by this death.

    Thus we find two problems already wrapped up in the remark about a transition from "is" to "ought"; now supposing that we had clarified the "relative bruteness" of facts on the one hand, and the notions involved in "needing," and "flourishing" on the other-there would still remain a third point. For, following Hume, someone might say: Perhaps you have made out your point about a transition from "is" This comment, it seems to me, would be correct. This word"ought," having become a word of mere mesmeric force, could not, in the character of having that force, be inferred from anything whatever. It may be objected that it could be inferred from other "morally ought" sentences: but that cannot be true. The appearance that this is so is produced by the fact that we say "All men are q" and "Socrates is a man" implies "Socrates is O." But here "O" is a dummy predicate. We mean that if you substitute a real predicate for "O" the implication is valid. A real predicate is required; not just a word containing no intelligible thought: a word retaining the suggestion of force, and apt to have a strong psychological effect, but which no longer signifies a real concept at all.

    This death is felt in the context of philosophical discourse by words like "ought" and concepts of "moral obligation" having a bizarre centralising character, a "mesmeric force", which stops these words from functioning as they do in every day moral decisions/actions/evaluations and thereby renders them irrelevant to practical concern and severed from any context that would render them intelligible.

    Edit: This is quite Wittgensteinian, philosophical "analysis" of morality has actually shifted the context in which (such notions of) morality made sense. "Language on holiday" kind of thing.

    The other part of the appeal to moral psychology in the essay appears to be to gesture towards analysing human needs and wants and human character with reference to our social practices. The cue for their analysis should be taken from how we talk and act and feel and want.

    But meanwhile-is it not clear that there are several concepts that need investigating simply as part of the philosophy of psychology and,-as I should recommend-banishing ethics totally from our minds? Namely-to begin with: "action," "intention," "pleasure," "wanting." More will probably turn up if we start with these. Eventually it might be possible to advance to considering the concept "virtue"; with which, I suppose, we should be beginning some sort of a study of ethics.
  • Anscombe's "Modern Moral Philosophy"
    You didnt read what I wrote. — frank

    More words, more thoroughly, then.
  • Anscombe's "Modern Moral Philosophy"
    You just repeated what I said. — frank

    If you can't see the distinction between appealing to social norms to set out what "moral and immoral" means vs analysing social situations to re-evaluate what the topic of moral philosophy should be I'd suggest you read the paper more closely.
  • Against Fideism
    It also might help, if you've not done so already, to have a very brief blurb before your epistemic theses on what you think an epistemology account should look like, as it stands it looks kinda via-negativa, though in the nice "vanquishing your enemies and assembling victory from their bones" sense rather than the theological one.
  • Against Fideism
    Unsure if you've talked about the distinction between reasons and causes so far. Reference to "for no reason" in the original paragraph is clearly intended to refer to a lacking (any) epistemic/justifying properties of the statement, but as @Wayfarer highlighted someone might rejoinder that there can be non-epistemic/justifying reasons for believing statements. (and these might be "unavoidable" or "well grounded" and bleed relativism into your project through inequivalent foundations, if you end up appealing to foundational knowledge items at the periphery of your 'chain of reasons').
  • Mathematicist Genesis
    Mathematical structures involving operations; like arithmetic; usually contain many copies of expressions which evaluate to the same thing. In this regard, it makes sense to identify expressions in the structure up to some equivalence relation. Like what was done above, expressions involving operations were identified up to the natural number arrived at by evaluating the expression.

    In general, this is phrased "X up to equivalence relation R", and is written X/R. You can think of equivalence relations as gluing terms together into meaningful chunks, for a prescribed sense of meaning set out in the definition of the equivalence relation.

    EG: "identify the natural numbers up to their remainder upon division by 2", the equivalence relation there is characterised as xRy iff x-y 's remainder upon division by 2 is 0. This gives you that 0 is related to all the even numbers and that 1 is related to all the odd numbers. This is called "the integers modulo (mod) 2".

    When you glue parts of a structure together, the glued parts tend to inherit the operations of the original structure. EG, two numbers: 3 and 7, 3+7 = 10. If you map this expression to their equivalence classes mod 2, [3] = [1], [7]=[1], [10]=[0], [3+7]=[3]+[7]=[1]+[1]=[1+1]=[2]=[0]=[10]. Everything works out.

    For the remainder of the discussion; whenever a structure with some operations on it (like natural numbers with addition etc) is introduced, assume that we have identified the terms of the structure up to equivalence of evaluating expressions. So for example, 22/7 will be identified with 44/14, or . Explicitly setting out how this works in every case is extremely tedious and unenlightening, so I'm just going to assume that once the structure has been defined the (collection of) equivalence relations upon its terms I just described the nature of have been set up appropriately. This means I will be abusing the equality symbol a lot, but the meaning should be clear from the context (and if I don't think it is I'll make a reminder).
  • Anscombe's "Modern Moral Philosophy"
    ↪StreetlightX


    I read this list as.

    Summary: we can defeasibily evaluate what context a statement is made in and what this evaluated structure requires for the statement to be true.

    (1) There is a range of sets of such descriptions xyz such that some set of the range must be true if the description A is to be true. But the range can only ever be roughly indicated, and the way to indicate it is by giving a few diverse examples. — Anscombe, 'On Brute Facts'

    We can describe the contexts that would make the statement "I owe the grocer 3 shillings" true, but not exhaustively and completely.

    (2) The existence of the description A in the language in which it occurs presupposes a context, which we will call "the institution behind A "; this context may or may not be presupposed to elements in the descriptions xyz. For example, the institution of buying and selling is presupposed to the description "sending a bill", as it is to "being owed for goods received", but not to the description "supplying potatoes ".

    ""The grocer provided me with goods" (for which I owe him 3 shillings)" - I could be provided with goods by other means than by the grocer, so "provided by goods" simpliciter is not part of the context of "I owe the grocer 3 shillings" but "The grocer provided me with goods now in the shop" would be.

    (3) A is not a description of the institution behind A.

    "The grocer provided me with goods" (for which I owe him 3 shillings) is not a description of what that requires to mean what it does. We don't have to read Adam Smith to buy groceries.

    (4) If some set holds out of the range of sets of descriptions some of which must hold if A is to hold, and if the institution behind A exists, then " in normal circumstances" A holds. The meaning of "in normal circumstances" can only be indicated roughly, by giving examples of exceptional circumstances
    in which A would not hold.

    Leveraging defeater examples to circumscribe the context. "I owe the grocer 3 shillings" would not be true if the groceries were North Korean Gettier identical plastic grocery copies and I was being conned.

    (5) To assert the truth of A is not to assert that the circumstances were "normal"; but if one is asked to justify A, the truth of the description xyz is in normal circumstances an adequate justification: A is not verified by any further facts.

    You can justify a claim by appealing to the context it's made in; as in, "this context is commensurate with the claim's usual function".

    (6) If A entails some other description B, then xyz cannot generally be said to entail B, but xyZ together with normality of circumstances relatively to such descriptions as A can be said to entail B

    Some type of type/token distinction thing? If A entails B, and we justify A with XYZ, we can only justify B with XYZ when we can be assured that XYZ is of an appropriate type. "I owe the grocer 3 shillings" => "I owe the grocer x dollars", but what if the grocer doesn't accept dollars? Would need to add "the grocer accepts dollars" to xyz for the implication to hold true in the context. A logical entailment of statements in XYZ does not entail that if the premise is conformable to standards XYZ then the conclusion is conformable to standards XYZ.
  • Anscombe's "Modern Moral Philosophy"
    As A points out, social norms obviously aren't the basis of morality (considering what those norms are apt to be.) — frank

    The paper doesn't appeal to social norms to vouchsafe what is moral or immoral (it goes against this use of moral and immoral explicitly). The paper uses social circumstances to ground ethical consideration in life (again), not in rules - be they abstract or social codes. It's trying to shift the territory of moral philosophy, not redraw the map.
  • Anscombe's "Modern Moral Philosophy"
    So it looks like there are two thrusts of argument in the paper, and a broad theme.

    Broad theme; why does "modern moral philosophy" concern itself so widely with the troubled notion of "morally ought"?

    (1) There's a historical account about Christianity and divine law theory in ethics. Ethical principles framed in terms of "I ought to do X" make sense because they draw upon a narrative background of "Thou shalt X". A broadly Aristotelian view of ethics (virtue ethics) is gestured towards as an alternative.

    (2) (a) Using Hume's is ought gap to pan for gold. A few times in the paper, a (caricature) of other moral philosophers' views have Hume's is-ought gap applied to them. The purpose of this is twofold; it buttresses the interpretation of "moral language" in (1) being consistent with divine law, as "I ought to do X" takes on a special (philosophical-juridical/divine) sense - this component is repudiative; if you buy this special sense, you've got the is ought gap to deal with, and Anscombe tries to describe precisely how the gap manifests once this special sense is bought. (b) The manifestations of the gap are then addressed, by rejecting the special sense and descending from the special sense of "morally ought" to psychological proclivity and worldly action; which Anscombe analyses through a mixture of appeal to ordinary language use and a gesture (of possibility) towards the virtue ethics referenced in (1). The second component simultaneously tries to dissolve the problem posed by the is-ought gap by rejecting how it frames moral judgements and "fills it in" with appeals to context; how stuff works and everyday word functions of "just" and so on.

    The second argument thrust is more technical, and appeals to a relation of facts to descriptions; collections of facts can be brute relative to a description. This is is the idea which is proposed to dissolve the is-ought problem and give a connection between psychological proclivity and worldly activity in Anscombe's new conception of moral philosophy.

    Hume's is ought problem can be stated as: truths are either relations of ideas (logic) or matters of fact (observation) [Hume's fork], an argument that goes from a statement of fact to an "I ought to do X" statement requires a justification for going from the statement of fact to the ought claim; but "ought" claims are neither observations nor deductive relations, and so always require an unjustified, separate premise on this basis.

    "brute relative to description" looks to be "brute relative to a context". As much as @Banno will protest to using the word "meaning", a rough characterisation of when a fact X is brute relative to a description Y is when the meaning of Y requires the truth of X in normal circumstances. EG, when the cashier tells you "That will be 3 shillings guv'" at the grocery when purchasing your shopping, "I must pay 3 shillings" is brute relative to "That will be 3 shillings guv" in the context of buying shopping in normal circumstances because "I must pay 3 shillings" is true in this circumstance. The separation of "ought" from "is" is replaced by the proximity of "owing" and "I must pay the cashier". Adequate understanding of what's going on allegedly dissolves the need for the "hidden premise" highlighted in the is-ought gap.

    The overall pressure this places on the is-ought gap is that certain facts must be assumed in order to do something in a context successfully. These facts can include moral language, like "owe" and "needs". The every day character of these moral items places a pressure on the prequisite crystalline perfection of divine law that Anscombe characterises "modern moral philosophy" as tasting of., We already have a basis from which to do moral philosophy; everyday life. Moral analysis begins in our socially saturated world not in the miracle of stone tablets or abstract principles.
  • Mathematicist Genesis
    Philosophical side note:

    Different senses of identity can be thought of as equivalence classes under different relations. Numerical identity is like 2=2, predicative identities like "crimson = scarlet" (insofar as they are both red colours) are equivalence relations of a different sort. This makes some sense so long as the idea can be understood extensionally without simplification. Something can be understood extensionally when the concept is completely expressible by collecting all terms that the concept applies to.

    EG: "foot " is an element of the equivalence class of nouns in English. Two words are related when and only when they are of the same grammatical type.
    EG: "foot" is an element of the equivalence class of "foot" under the equivalence relation of string (character sequence) identity.

    Not all concepts behave extensionally.

    EG: "pile of rocks" has ambiguous boundaries, at what point does a collection of stones become a pile?
    EG: "hot". at what point does a temperature become hot? Does a temperature become hot in the same way as a human body feels hot?

    Expressions in natural language rarely only leverage extensional concepts to furnish their understanding, most expressions are confused jumbles. This provides lots of interpretive issues, but what we lose in rigidity and precision we gain in expressive power.
  • Mathematicist Genesis
    The previous post talked about equations and how to define the algebra of arithmetic. It turned out that the algebra of arithmetic invokes a notion of equivalent expressions; two expressions are equivalent when they are equal to the same number. 1+1=2, 3-1=2, therefore 1+1=3-1. This idea of equivalence is so general that it should be spoken about in detail; there is a general formalism for it called an equivalence relation.

    So what's a relation? A relation on a set is a collection of pairs of elements from that set where the order matters. EG the set {1,2,3} can have the relation R={ (1,2), (2,3) } on it. This says that "1 is related to 2" and "2 is related to 3". The pairs can be written as 1R2, 2R3. Note that 1R2 is different from 2R1! An equivalence relation has three properties which characterise it:

    (1) Reflexivity: for all x; xRx
    (2) Symmetry: for all x for all y : xRy implies yRx
    (3) Transitivity: for all x for all y for all z : xRy & yRz imply xRz.

    The relation defined above satisfies these properties.

    (1) Reflexivity: eg; 3+1 = 3+1
    (2) Symmetry; eg; 3+1 = 4 implies 4 = 3+1
    (3) Transitivity; 3+1 = 4 and 4 = 5-1 implies 3+1 = 5-1

    Equivalence relations defined on some set divide that set up into non-overlapping parts. These are called equivalence classes. The equivalence class of x is written [x]. EG [4] contains 3+1 and 5-1. But it does not contain 3, or anything that adds up to 3, like 2+1. The equivalence classes on a set divide it up into distinct parts which are somehow equivalent. The above construction divides the set of expressions of the form x+y up into the various sets [t]={x+y = t}.

    Note that [2] = [1+1]. When an equivalence class can be written in the form [x], x is called a representative of that equivalence class.

    The properties of equivalence classes (for arithmetic) exhibited above give you [x]+[y]=[x+y]. The same applies for every way of obtaining x from arithmetic operations on any collection of numbers. This means that the arithmetic operations defined on numbers transfer to arithmetic operations defined on equivalence classes. Numerical equality of what arithmetic expressions evaluate to (3+1 evaluates to 4) transfers to set equality of these equivalence classes ([3+1]=[4]). Once you get arithmetic expressions evaluating to terms functionally (which they always will, they will have unique outputs per input set by definition), you can always leverage this to define equivalence classes of expressions on whatever structure you have defined the operations on.

    This technique is used over and over again; once operations are defined on a structure, equivalence classes of evaluation (like 3+1 evaluates to 4) can be used to motivate algebraic substitution/solving things by algebra on that structure. The same will apply to rational numbers (fractions); they will be built as equivalence classes of naturals. Then real numbers will be built as equivalence classes of fractions!
  • My own (personal) beef with the real numbers
    Here's an argument for the necessity of the empty set.

    If you want a theory of sets, you want to be able to compare pairs of sets. You want to know if they have any elements in common, and aggregate all elements they have in common into some set at the same time; taking an intersection of sets.

    Let's compare the interval of real numbers (1,2) and the interval of real numbers (3,4). Assume that the intersection of those sets must be non-empty, then there is is a number which is both strictly less than two and strictly greater than three. There is no such number.

    The empty set comes quite naturally from two principles:
    (1) The ability to state what elements sets have in common.
    (1a) The elements that sets have in common must always be equal to a set.
    (2) That two sets might be disjoint.

    (as @fishfry pointed out, you can get contradictions if you disallow an empty set like object)

    One possibility, seemingly proposed by @Mephist (if I understood your responses to MU anyway) is that you put in the empty set as a proper class primitive into the theory, so that (1a) is denied but (1) and (2) are still true. It's pretty much a distinction without a difference though; "intersections of sets contain at least 1 element or are the empty class", I'd imagine you'd have transfer results from this "empty set is declared a proper class for no reason + ZFC" theory to the usual ZFC theory. My guess is that it would be a distinction without a difference.
  • My own (personal) beef with the real numbers
    Yes, however in my opinion Anders Kock's book ( https://users-math.au.dk/~kock/sdg99.pdf ) is not so difficult to understand. d in my opinion should not be interpreted as a number, but as the base of a vector space made of infinitesimal numbers attached to each of the real numbers of a "base" space — Mephist

    You get something really similar to that with any mapping , where is some manifold in which is a point. With the constraint that , the collection of all such maps forms a module ("vector space with elements from a ring"). It's an infinitesimal tangent space attached to the point. It might mutilate intuitions of the real number line, but that doesn't matter, as it seems designed to simplify language and proofs about smooth functions. Whether it's "wrong" or not is just a question of taste and application.

    Can you show me a proof of consistency of ZFC set theory that doesn't make use of another even more complex and convoluted set theory? — Mephist


    As I'm sure you know, if a theory's consistent and has arithmetic, it can't prove its own consistency. You always have to go outside a theory to prove that theory's consistency; so consistency of system X is always just relative consistency to some other system Y. If you want to find a model of some axiom system, you need to construct the model through other rules (even if they're incredibly similar).

    A weaker criterion might be that you want some theory which is equivalent to ZFC in the sense of having statements which are inter-derivable with all of its axioms. But then all consistency proofs here would establish are that "equivalent axiomatisations of ZFC are consistent at the same time"; like being reassured that ZFC is consistent because you can replace the C with Zorn's Lemma and that ZF+(Zorn's Lemma) is consistent when ZFC is.

    I'd've thought you'd be quite happy with small categories? :brow: Aren't the models of intuitionistic logic Heyting algebras (from earlier) anyway? They're sets, or categories which are represent-able as sets. So I'm reading this like: "I don't like sets because I don't like the structures that establish ZFC has models. But I like intuitionist dependent type theory because it has a model! (Which is a collection of sets.)"
  • Thoughts on love versus being "in love"
    And what would humans be without love?"
    RARE, said Death.     — Terry Pratchett, Sourcery
  • My own (personal) beef with the real numbers
    I don't really understand this. — Mephist

    It seems to let you rigorously think of dx as an infinitely small translation/length with the expected connections to calculus and differential geometry.
  • My own (personal) beef with the real numbers
    ↪Mephist


    Page 4. Properties.

    "However if xd = 8d for all d then x=8". In that link, they are multiplying by d.
  • My own (personal) beef with the real numbers
    ↪Mephist


    All you need to do to turn any non-zero dividing element into a zero dividing element is to multiply it by d. So all elements zero divide. If you quotient out the infinitesimals by mapping to standard parts it's a field again, as it's just the reals. Zero divisors can't have multiplicative inverses.

    If you have two nonzero ring elements a and b, and a.b = 0, assume wlog that b has an inverse, then a.b.b^-1 = 0.b^-1, then a=0, so b has no inverse.
  • My own (personal) beef with the real numbers
    The real numbers — Mephist

    The real numbers are a field (without 0). The real numbers augmented with the set of numbers defined in the previous post's link are not.
  • My own (personal) beef with the real numbers
    You cannot divide by a number that has it's "base" part 0. That works at the same way as the usual real numbers — Mephist

    Ok. Assume d had a multiplicative inverse. Also d^2=0. Then d.d^-1=0.d^-1. Which gives d=0. But d is nonzero. So d does not have a multiplicative inverse. All nonzero elements need a multiplicative inverse for it to be a field.
  • My own (personal) beef with the real numbers
    Why not? Which of the field axioms are not satisfied? — Mephist

    Multiplicative inverses, you have zero divisors from the infinitesimals and that blocks it.

    Why is this a problem? — Mephist

    It's not, it's just a point of difference. Having a number system that's built to embed differential geometry intuitions and make differential equation diagrams rigorous is very cool.
  • My own (personal) beef with the real numbers
    The fact that you have to use intuitionistic logic with it's weird rules about double negation, to make sense of "d^2 = 0" even if "d =/= 0" in my opinion is just a "wrong" definition of the negation operator: it should be called "complement of" instead of "not" (for example we should say "d belongs to the complement of 0" instead of "d =/= 0"). — Mephist

    "true continuity" can be defined even using standard set theory. Actually, even category theory can (and usually is) be based on standard set theory. — Mephist

    It's not really about the reals and analysis as usually thought of, the construction doesn't satisfy the field axioms. Moreover, every function from the constructed number line with infinitesimals to itself turns out to be smooth (so, continuous and differentiable). If I've read right anyway.
  • My own (personal) beef with the real numbers
    ↪Mephist


    I'm only pointing out that if sets can't provide a model for @aletheist 's intuition of continuity, since they consist of distinct entities (you can distinguish infinitesimals from each other, and make claims like lays in some set), then neither can the synthetic axioms presented in that link, as they concern sets.

    (Though, they can be presented in a more general context. What I'm pointing out is that while sets aren't necessary to talk about continuity in that sense, they don't preclude it by themselves either.)
  • My own (personal) beef with the real numbers
    Exactly. This is a sheaf of linear tangent spaces built on the base space of Cauchy sequences of rational numbers. Similar to the sheaf of all vector spaces tangent to a sphere. — Mephist

    Still sets tho. .
  • What makes a government “small”?
    3) Some human desires are amoral or immoral.
    4) Therefore, market forces are informed, in part, by an amoral or immoral element     — ZzzoneiroCosm

    Logical fallacy. It is still possible that all market forces are informed only by moral desires! Praise Xenu!
  • Brexit
    Think it's about time to convert to Nick Land, the answer is always to make things worse faster.
  • Mathematicist Genesis
    There's sort of a branching point here, @Pfhorrest, two ways to tell the same story. Now we've got resources to talk about sets, operations on sets and so forth, we could go through the specific cases of natural number arithmetic -> fractions -> real numbers, building fractions from sets of natural numbers, building real numbers from sets of fractions, or alternatively we could go into abstract algebra and talk about the same things in terms of semigroups, groups, rings and fields.

    The first has the benefit of telling a story from the "objects" points of view, how do we build up more complicated objects from simpler objects, the second has the benefit of telling the story from the "structures" point of view, which presents the integers as "solving a problem regarding addition and subtraction in the naturals", the fractions as "solving a problem regarding multiplication and division in the integers" and the real numbers as "solving a problem regarding sequences in the fractions".

    Input?
  • Mathematicist Genesis
    Ideally what is needed, given the equation concept, is a way to manipulate equations so that their solutions can be revealed. What this requires is setting up an idea of the substitution of expressions in the same variable which are true whenever the original equation is true.

    EG: is true implies is true implies is true implies is a solution of .

    What we'd need is a way of substituting equivalent expressions for equivalent expressions, and in order to do this we can define what it means for two expressions to be equivalent.

    We'd really like something like , so that all ways of obtaining from operations of arithmetic are in some way equivalent. This can be set up for addition as follows:



    which is to be read "the set of all expressions defined as such that every expression within is the sum of two numbers which equal .

    If we took some other number, say , and defined the same thing:

    We could ask "under what conditions is ?", the answer there is:



    Under what conditions does this equation of sets have a (non-empty collection of) solutions? Well, whenever . That is, whenever the expressions in the two sets equal the same number. This gives us a perspective from which equations can be manipulated.

    Notice that if we define , this also respects the above. (For example, any expressions which evaluate to 2 plus any expressions which evaluate to 3 equal any expressions which evaluate to 5, this is 2+3=5 if and only if [2]+[3]=[5]).

    If we have that , then we also have that , this means we can substitute in anything in for in the equation and obtain another equivalent equation. Substituting equivalent expressions is how equations can be solved through algebra. Imagine that we've done the same thing for subtraction and so on, eg:



    For example:


    What this highlights is that you can leverage the equality defined between sets to define an equivalence between expressions that equal the same constant. This (kind of thing, but for all the operations) is the construction that formalises the algebra used in solving equations.
  • Mathematicist Genesis
    One important technique is how to construct and solve equations, and we've got the theoretical resources in place to do that for natural numbers arithmetic. So what is an equation? And what is a solution for an equation?

    An equation in a variable in a structure (with defined) is an expression of the form , and where is a function defined in the structure from its constants to its constants and is a constant element. A solution for a given equation is a substitution of for a constant term in the equation expression which makes true.

    EG: , substituted into the above formula gives , and a solution is then since the substitution makes true.
  • Mathematicist Genesis
    There are other axioms of ZFC. But considering that the story being told is "how to go from formal languages to differential equations", I'm not going to go through them. The axiom of infinity and the successor function give you the natural numbers, the axiom of extensionality gives you equality between natural numbers when defined in that way.

    The strategy for showing that ZFC contains within it a copy of the natural numbers and arithmetic is to show that you can define a collection of sets and the relevant operations/functions , (and subtraction, multiplication, division, inequalities on natural numbers etc etc...) such that the overall structure satisfies an axiomatisation of arithmetic like (first order) Peano arithmetic .

    What this does formally is show that (the set theory construction) models (the Peano arithmetic axioms defining natural numbers and addition). The successor in Peano arithmetic is interpreted as the successor in the set theory construction, the numbers in Peano arithmetic are interpreted by their corresponding number in the set theory construction, addition in Peano arithmetic is interpreted by addition in the set theory construction and so on for however many functions/operations/relations you throw into the Peano arithmetic structure and the set theory structure.

    Recapitulating a previous point: the power of set theory is that it provides a way of defining lots of structures, so that you can take some other structure (like Peano arithmetic) and interpret it as a set theory object (collection of sets).
  • Mathematicist Genesis
    Or alternatively "take the thing on the left, and successor function it the number of times required to reach the thing on the right from 0", equivalently "union the thing on the left with the successor function applied to it as many times as the number on the right" — fdrake

  • Mathematicist Genesis
    "take the thing on the left and write it as its set, take the thing on the right and write it as its set, define the sum as the successor function applied to 0 as many times as the thing on the left plus the thing on the right" — fdrake

    Or alternatively "take the thing on the left, and successor function it the number of times required to reach the thing on the right from 0", equivalently "union the thing on the left with the successor function applied to it as many times as the number on the right"
  • Mathematicist Genesis
    An interesting property of the set as defined above is the induction property. It allows a kind of proof called inductive proof. This provides a way of proving things about sets of natural numbers. Induction works like:

    (1) Show that some statement is true of some number
    (2) Show that if is true of , then is true of .
    (3) Conclude: therefore is true of for all .

    EG: Theorem: "Given a sequence for in , show that the sum of the first terms of this sequence is .

    Proof:
    (1) Initial statement: then .

    Now how do we show (2)? We can use the strategy of conditional proof detailed earlier, simply assume that is true and then show that is true on this basis. "Assuming is true" is called the induction hypothesis. So we make that induction hypothesis.

    Assume , then is , which is , which after some tedious algebra is .

    (3) The combination of (1) and (2) shows that this is true for all the numbers.

    Is it possible to write "1 + 1 = 2" entirely in terms of sets and set operations? Like, if I understand correctly, we can say that 0 = ∅ = {}, and that 1 = {∅} = {{}}, no? So 2 = {{},{{}}}, right? — Pfhorrest

    Yes. The definition of addition lets you do this, it's just an exercise of translation.

    and . Gives you that if is a set which represents a natural number and so is , . Working out the details for is , working out the details for is . Defining without the inductive definition gives you "take the thing on the left and write it as its set, take the thing on the right and write it as its set, define the sum as the successor function applied to 0 as many times as the thing on the left plus the thing on the right".
  • Mathematicist Genesis
    Now onto the axiom of infinity.

    The axiom of infinity states that:

    "There is a set such that is in , and if the set is in then is in ". Where is defined as .

    This looks like:



    If we define the symbols , ,
    ... ,

    we can see that looks suspiciously like the natural numbers. This is true. It is the natural numbers. Arithmetic can be defined on this set by defining an operation ( a function from to ) which is given by:




    This lets us prove in .

    Theorem:
    Proof: Let be in , then . Then set , this gives .
  • Absolute rest is impossible - All is motion
    ↪frank


    Ok sorry. Now I'm back to no idea of what you're talking about.
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