When he joined, he was suspected of being Jake due to posting style and post content. He wasn't banned immediately because there wasn't sufficient evidence that he was Jake. Now, he's claimed to be a returning banned member in a PM.

Suicide by mod kinda thing.

Okay. The period of the video and the video itself is about globalism. Globalism may have pulled some people out of poverty, which wouldn’t have been hard, but it then placed other people in trouble. I don’t think I need to explain all that. In that sense globalism is a big con, a global con job. We know who benefitted. But historically, going back as far as the industrial revolution, capitalism has slowly drawn people out of crippling poverty. There’s no doubt it created problems. But no other system has done this, until now, which is China. China is not Russia, they’re flexible with their Marxist ideology, they’re pragmatic. Which is interesting. — Brett

@Hippyhead was banned for being a returned banned member.

My friend, I had no such illusions going in. But please don't discount the possibility that someone other than you and I, reading this, might benefit from seeing a lefty twit get handed his arse over and over again. — counterpunch

:up:

If this imaginary audience wants to read a more civilised and in depth discussion of related issues; which includes citations; I invite them to read here.

A disposition towards some state of affairs: this disposition presents itself as a conditional "if...then.." which is a pragmatic construction. Dispositions are anticipatory and language and logic merely formalizes this. — Constance

I'd like to say that I generally agree with your pragmatic account of dispositions, and I think we have similar inspirations for holding similar accounts; exposure to Heidegger, going by your reference to "ready-to-hand". I think where we may lose agreement (though I dunno), is whether the act of putting something into words acts as a transformation of semantic content. If we admit that pragmatic engagements have their own broader flavour of semantic content than the semantic content of declarative sentences, and we take a pragmatic (speech-act centric) view on the use of language, it would seem that the broader flavour of semantic content is generic in the use of language, not simply the kind of content which we use declarative sentences to express through their connection to truthmakers/truth conditions.

I don't mean to suggest that "for every event E possibly there exists a sentence S(E) such that E is the truth maker for S(E)" is strictly false; I think it describes one way of using language rather than a property of nature. If we attempt to describe any phenomenon, it will tend to be cashed out in part using declarative sentences; but it will also contain allusions, metaphors, narrative and rhetorical devices, which function more by cajoling (performativity; illucutionary force and the expectation of perlocutionary effect) than by matching up with a truth maker. This second cajoling kind of content I take as corresponding to a part of the broader conceptualisation of semantic content above. It isn't spelled out in what makes an utterance true, it can only be spelled out in terms of its expected effects and motivations.

To be sure, those expected effects and motivations can in principle be stated as occurrent afterwards, but that switches from a kind of semantic content that is fleshed out by truth conditions to one which is fleshed out by pragmatic considerations.

I quote the passage above to illustrate where we came in on this question - just a day or so ago, and how already, the point has wandered quite a ways from its origins. If it weren't possible to click back a page or two, and look up where we came in - I would be quite lost. I really couldn't explain why we are seeking to establish the precise mildness of your approval for removing statues that remind us where we came from. — counterpunch

Points tend to multiply when following them along their lines of connection. To my reckoning, you accused me of being inconsistent when I said I was indifferent to statues but approved of those statues being torn down. I replied describing that I had a mild preference for them being torn down.

I'd initially brought up the statues because I believed it would be an issue which shows that some political issues aren't resolvable by scientific means, and I believed you'd disagree with me on whether tearing down those statues was permissible.

I do worry though.

"The world's two largest standing Buddhas - one of them 165ft high - were blown up by the Taliban in Afghanistan at the weekend. After failing to destroy the 1,700-year-old sandstone statues of Buddha with anti-aircraft and tank fire, the Taliban brought a lorryload of dynamite from Kabul."

How mild is your approval for this? Or do you disapprove of this - and maintain it's only your lefty cultural vandalism that's praiseworthy? — counterpunch

My approval/disapproval for tearing down a statue depends on what it represents, why it was torn down, and what the act enables. I don't know enough about that act of vandalism to form much of an opinion about it.

You don't? Ancient Egyptians, Greek, Romans all had slaves did they not? Ottomans, Muslims, Africans, Russians all had slaves. British people were slaves until 1584; only they called them serfs. Slavery is the default, and capitalism is the cure. Don't be sly - making sideways arguments, and referencing books I haven't read, and am obviously not about to run out and buy. Slavery was everywhere - all around the world and throughout all of history until the West ended it. — counterpunch

I think a book reference is rather generous actually. If you're not going to check reputable contrary citations to your historical+anthropological claims, I don't think there's much point talking about the issue with you.

Another sly argument. In society and economics, it's necessary to discriminate - for example, between people who are qualified for a job, and those who are not qualified. So, for example, if numerous black people applied for a job without having the necessary qualifications, by your logic - they are being discriminated against, relative to the white person who is qualified. The discrimination isn't racial discrimination, but you switch effect with cause - like with Redlining, to suggest a racial disparity in effect proves racist intent as a cause. It's not so. That's politically correct logic. The same logic that denies slavery existed everywhere, since the dawn of time. You - lefties, are not capable of an honest argument. — counterpunch

I gave you an easily attackable candidate definition/clarification of how I understand systemic discrimination. If you are unwilling to engage with the term on its terms, again, I don't think either of us will benefit from continuing the discussion.

Prelinguistic? Pragmatics is this, and most of our engagements in the world are like this. — Constance

I'd just add that each step of the process can be put into the form "Constance believes P" where P is some proposition. — Banno

Do either of you see a tension between "most of our engagements with the world are (prelinguistic)" and "I agree, and those engagements target statements"?

Isn't your complaint just that I do not say enough about that of which we cannot speak? — Banno

My complaint is that speaking about that which we allegedly cannot speak is a routine function of language. The stuff we allegedly can't speak about is already baked into use because use is an interaction with the world.

Oh, yeah, all that. Except "The content of what is said is what it shows" - "content" is wrong, as shown in PI - use replaces content. — Banno

I did not mean to suggest the content was "inside" the word, far from it. This is part of what puzzles me about your position, everything's sufficiently (semantically) externalised that it's functionality all the way down in terms of meaning, except for how declarative sentences play a privileged role in the articulation of mental content (or "mental function" if you prefer) and events, those statements are "inside" those dispositional states and events in the same way. It's like you've taken Wittgenstein's propositional glasses off like he does in the PI but put them back on again when reading something else! They're like analytic philosophy reading glasses.

Fair enough.

There is a difference between saying and showing.
Saying is a kind of showing.
What is said will align with a picture of the world only when its frame (true/false) is held up.
Holding up the frame limits what is shown to the frame of what is said.
The content of what is said is what it shows.
Showing puts the picture in place and paints it beforehand.

My argument: Read the Tractatus and associated material — Banno

I guess you're done here, then.

You seem to be equivocating on different senses of what it means to say that events could potentially be stated. It could be said that, at a time when there were no language users, events could not potentially be stated (because there were no language users), but it could equally be said that those events could potentially be stated (because there could potentially have been language users). The first is an expression of actual or real potential, and the second is an expression of purely logical potential. — Janus

Those aren't my equivocations, they're attempts to flesh out what "can" means in "Every event can be stated"! Since my discussion partners don't seem to want to flesh out the modality associated with it, despite making a modal claim, I decided to try it.

The world is all that is the case. — Banno

Argue for it?

The kind of people who will only “keep in touch” by adding you on facebook are precisely the kind of people it’s not worth the effort of keeping in touch with. — Pfhorrest

I think you're trivialising those friendships that I had; we knew each other for between 6 and 11 years, regularly saw each other in person, helped each other out etc. Duration depending on the person. In general I agree with you though!

Yes, an expectation of the reader filling in the gaps is common. It would be dreadful if that practice were abandoned! :cool: — jgill

Yes! I don't think mathematical research would look anything like it does if we had to reference all conventions we're using. What that does do, however, is make mathematical demonstrations heuristic in Lakatos' sense; more about displaying the concepts to a sufficient degree of obviousness than mandating that all proofs have every step of reasoning spelled out symbolically.

I think that "sufficient degree" has been amplified since the "crisis in analysis"; the whole thing about people not distinguishing continuity from the intermediate value property and various problems I can't remember with integrals; and amplified again since the Hilbert program and the invention of formal languages as mathematical abstractions.

You're indifferent to statues, but support tearing them down. Yeah, that makes sense. — counterpunch

I don't think it changes much, but is a mildly good thing that they're torn down. Mild preference for them being torn down. I'd have a stronger preference if I saw stronger evidence that tearing them down helped fight systemic discrimination, I'd have a weaker preference if I saw evidence that it doesn't help at all. I haven't seen any evidence, or convincing arguments, that tearing down the statues harms people or society. I've seen some evidence, and convincing arguments, that it helps in a limited way.

I used the term "you" in a collective sense - meaning, you left wing types. It came across as personal to you, and was more aggressive in tone than I intended. For that I apologise — counterpunch

:up:

No worries, I have the same bad habits of stereotyping the people I seem to disagree with.

This obvious left wing hypocrisy can only be achieved by denying the ubiquity of slavery to the human condition, and that's not a lesson you want to forget. Those statues should make you aware, and happy you live in a society and era that allows for individual liberty. Yet you, lefties - protest against the very society, philosophies and economic system that afford you that freedom. You act as if freedom is some natural default setting. It's not. Slavery is the default - and freedom is hard won. So no, it's not quite: — counterpunch

I don't think "slavery is the default" fits the anthropological record; from what I've read of it there's very few things that compare to international slave raids and barter of people between societies. David Graeber's "Debt, The First 5000 years" has rather a lot of relevant detail on this. I wouldn't say this automatically means anything like modern freedom fits the record either; we're kinda similar to Ancient Greek slaves - we form contracts with people for performing services and receive money as payment.

t's that you falsify the history to forge a weapon against the very system that affords you the freedom to have an opinion, and to express it. You turn our own achievements against us - and it's just dishonest. We have multi-cultural societies, and have set racial equality in law, but it's still not good enough for you lefties - because, when it comes down to it, you're playing identity politics, and it's a power game. It's you lefties stirring up racial animosity for political advantage - not the right. Western society isn't institutionally racist, there isn't a racist genocide being committed by the police. It's all a lie. You politically correct lefties are the real racists, only you're racist against white people so that's okay then! . — counterpunch

Formal equality under the law doesn't signal the absence of systemic discrimination in a society. If you understand systemic discrimination as "institutional discrimination", I think you've been interpreting systemic discrimination in a limited way; missing the intended interpretation. It's more of a functional thing than a legal thing - closer to something like "a society can be said to be systemically discriminatory against group X iff belonging in group X amplifies exposure to negative outcomes relative to those who are not in group X AND that exposure has strong social+economic contributory causes". In general now we're talking about stuff like black people in the US being exposed more to dangerous levels of lead paint due to Jim Crow laws, redlining etc than a formalised legal difference between the groups. The same kinda thing for hiring disparities with equivalent CVs but you change the racialisation or gender association of the name! Formal inequality under the law (a form of institutionalised discrimination) would also fall under that general characterisation.

How does something that isn't there, say anything? You look at that statue of Colston, for example - and are offended by it. It means something different to you now than it meant to those who erected it, and it will mean something different again to subsequent generations. Who the giddy fuck are you to insist your current opinion, not only trumps that of previous generations, but removes it from the consideration of all subsequent generations? — counterpunch

I'm not offended by statues. I'm simply indifferent to them. My perspective is tearing them down is the best use of them, really. But I also don't expect tearing them down to do very much; at best it's a symbolic gesture, but I imagine that it sends a message as symbols do. Those statues were put up to reproduce history in a symbolic register, the mob which tears them down has the same function.

I believe the argument you've made applies to any action which goes against a tradition, regardless of the value of the tradition. I do side with you that traditions can be there for a reason, but I do think that their values should be weighed and measured. What those statues represent has been, and found wanting.

I reserve the right to judge traditions, just as you've done with the present state of the world and its habits of not listening to scientists! Ought that change? Who're you to go against the traditional relationship of science and politics? The argument you've used is very much a two edged blade.

What do you see? You see an anti western target for your politically correct virtue signalling - a myopic, self righteous view based on the lie that slavery was a particular cruelty invented and practiced by white people, against black people - because they're black. You think slavery was racism. But white people didn't go out and capture black people and force them into slavery. They bought slaves from black people; trading Western manufactured goods - cloth and metal tools for slaves, then trading salves for sugar and spices in the Americas, and then back to Europe to sell the sugar and spices. — counterpunch

I don't really like that you've told me what I see. What I see is the historical contingency that nations which largely consisted of people with white or light tan skin, for mercantile+economic reasons, decided that chattel slavery, indentured servitude and colonial expansion were good business ideas - these were the ur form of systemic racism. And then further historical contingencies that lead those slavers and their political ilk creating justifying ideologies for slavery - those are expressedly racist. We live in the legacy of those decisions, and their continued effects have been demonstrated beyond any reasonable doubt.

Slavery, structurally, clearly doesn't have to be a white person enslaving a nonwhite person. Slavery, insofar as it is relevant to our societies' present functions, absolutely was.

You despise the very civilisation that ended slavery — counterpunch

Nah man, I've got nothing but respect for Dessalines, and anyone that works to end slavery, apartheid, and discrimination of any sort.

and if you have your anti western way

Anti-western? I doubt you mean that in the sense that the west couldn't function as it does without systemic racism; with that I agree. But I don't really know why you would think I'm anti-western. What does that mean to you?

Formal mathematics is simply that employed by research math people these days. You know all this of course. I've now read a bit about the topic here and I suppose what I have used might be called naive heuristics, in light of all the various types of heuristics described in Wiki. — jgill

I think "formal mathematics" as the book considers is quite similar to "formal mathematics" as practicing mathematicians consider it. There is an aspect of "formal mathematics" - really a philosophical construal of mathematics- that Lakatos strongly criticises, though.

In my experience, practicing mathematicians simply don't expect everything they say or think to already be formalised, nor do they expect a strong mathematical argument to be presented with the required rigour of logic. As an example, this proof is logically invalid as presented but demonstrates what it seeks to to the point of obviousness.

Let $x$ be a real number that satisfies $x^2-1=0$.

$\begin{align}x^2 - 1 = 0\\\implies (x+1)(x-1)=0\\\implies x=\pm 1\end{align}$

It's definitely a true series of implications, but there's no supplementary statement that the factorisation in step 1 is valid , or a statement of the premise that the real numbers as a field have no zero-divisors. To be sure, it clearly demonstrates an understanding of how to solve the quadratic, and how you'd go about doing it, but it's not strictly a valid argument. Those things I highlighted which restore its logical validity are in the background understanding of the terms, and the reader is expected to grok it.

The heuristic analysis of Lakatos is a historical investigation into that kind of background. The book has a historically detailed example of how different background understandings of the concept of polyhedron played out in mathematics. Being able to take a concept out of that background; like polyhedron; and formalise it with precision is an advance in mathematics.

No one should have to square themselves with being a hermit in order to avoid using a website.

I agree that keeping in contact with people is still very much a useful thing. Now I simply prefer to use their Messenger application. The rest ("news" feed mainly, and the advertisements they bring, plus groups -- most of which are toxic) has become quite useless. I suggest this as a compromise to you if you're regretting not using it. It's basically text or, if people remember back, AIM (AOL Instant Messenger). — Xtrix

On the whole I don't regret not using it. I'd join again if I had a good use for it; like marketing or organisation, I'd put aside personal quibbles for that. At this point I think anyone I could actually socialise with... I'm already in regular contact with; the other dinosaurs and hermits I know. The thing about the social commons being monopolised is that it housing the commons disincentivizes forming another space; since it doesn't "house the commons" until it's reached a critical point of development, before that it's just a worse interface to social life. I'd have contact with the people I already have contact with, but not the other features of social life; so long as they're not represented on the less used service!

For sure, the bases of moderation and suspension/banning should be explicit in those T&C's. It is frustrating when platforms augment this with unofficial, ad hoc moderation (the Guardian being a prime example). But even if they don't, yes it's unfair, so what? That's a judgement on those platforms. As Pfhorrest said, simple solution is don't use them if you don't like them. — Kenosha Kid

@Hippyhead

Oh, I don't use them, and I regret it at least once a week. Here are some social things that happened since I stopped using social media:

(1) Many lifelong friends no longer contact me.
(2) I no longer get invited to any social gatherings I don't organise myself.

I've tried keeping in contact with friends and social groups; emails, phonecalls, Whatsapp; those are a relatively small window into their lives in terms of engagement-hours. Eventually people just stopped responding. I haven't had a regular group of friends, or been able to get one, since "taking my business elsewhere". Since "elsewhere" from social media is like being deleted from the commons. My presence is largely filtered out from the construction of social reality.

You may think it doesn't limit social interactions once they're already rolling; and for that I cite you the countless times someone has wanted to add me on Facebook, instead I offer them my email or phonenumber or Whatsapp, and the person immediately stops talking to me or never contacts me again using my suggested media. The intuition: what kind of weirdo exempts themselves from society like this?

I think perhaps you're both showing your age.

Looking back from that high energy sustainable future, yes - I think people would regret destroying history for what it symbolises today. I think providing the world with limitless clean energy from magma, securing the future for humankind makes good on the civilisation we fought to build, and that sanitising history removes a warning label from what might come again if we don't keep building. — counterpunch

Eh, those statues have sat there being a celebration of history for a long time, removing them is a symbolic act to acknowledge the ongoing relevance of the warning they represent.

In terms of "big picture" stuff we probably agree; if the world had limitless clean energy that everyone could access, our greatest existential threats would be solved. Getting there will be messy political business though, and I'd guess we'd disagree on what kind of politics is required to get there. Our lords and masters have a habit of doing whatever they can to limit our access to that future, since they're invested in keeping the material conditions in place that cause those existential threats and stifle us from doing anything about it.

I completely agree. It seems quite ridiculous to me to compare the T&C's of services provided by private interests to state censorship, especially as these complaints come from users of sites who would ban criticism of far-right violence without a second thought. If Twitter doesn't want formentors of violent coups on its user list, their house, their rules. — Kenosha Kid

So I agree that legally it looks like there's no free speech violations, since the platform has power to remove whatever content they like. There is a rational kernel to the free speech argument though. Large social media sites effectively function as the social commons; they're how we chat, make friends, inform ourselves and so on. It is quite creepy that someone can be exiled from that commons with little to no oversight.

I think the free speech complaint "goes through" so to speak, but not in the terms it's originally articulated in.

If the purported event is representable in language and it meets the public usage criteria for an event, then it is an event. — Andrew M

It strikes me that if there are a class of things that are "purported events", that must be "representable in language" and "meet public usage criteria" to count as events, it would then follow that those purported events do not fall under DPC despite being occurrent:

If DPC is: for all events E possibly there exists a statement S(E) such that E is the truth maker for S(E).

So long as there are "purported events" which are occurrent, there's a sneaky domain restriction on first quantifier; there are occurrent things which never have statements that state them in DPC's sense since they don't count as events. It makes what it means for something to happen depend on language use. Like the world is language's shadow.

After all, our conditional logical form very likely is constructed on t he foundation that pragmatically "mirrors" the primitive, non symbolic cat knowing. — Constance

:up:

I'm not trying to commit myself strongly to the thesis that "for every event E possibly there exists a statement S(E) such that E is the truth maker for S(E)." is false, I'm trying to point out that it requires an account.

The debate (first post here: ) did touch the issue of animal beliefs, my reading was that @creativesoul was criticising the following claim, which he saw as a consequence of @Banno's account: animal beliefs target statements because we can set some representation of them out using a statement. Banno in turn seemed to insist that all there was to the animal's belief was our attribution of a statement of that belief which is held true by the animal.

For my part, prior to the debate I asked both participants to make the motion more specific:

So, to save it from being Banno arguing that propositional content is a property of statements (or more generally speech acts) and since belief is a propositional attitude, the content of the belief is the proposition it's directed towards, and so concluding it must be propositional content.

And you arguing that belief content is a broader semantic category - I don't know what kind of things you throw in it, other than that it can be "pre-linguistic" - and so since not all of that content is even "linguistic" (presumably not all words or symbols, I don't know where you come from on this), not all of that content can be propositional; since propositions must be linguistic.

If you continued like that, Banno could assert his definition of belief, you could assert your definition of belief, and there's a strong chance you'll both address none of the other's points and retreat to hedges — fdrake

And I was dismissed, only for it to be unproductive in the way I specified. The real meat of the issue, I think, would've been to discuss something like:

Beliefs as mental states/dispositions with content vs beliefs as holding some statement to be true. Issues there might be: is a disposition towards a state of affairs the same as an attitude towards a statement? What role do t-sentences play in that account?

No. I don't propose drilling to the core of the earth. I was explaining how vast the energy of the earth is - 4000 miles deep, 26000 miles around. We could tap that energy forever and never put a dent in it. I suggest drilling close to magma chambers, and at subduction zones, where one continental plate meets another. There are about 500 volcanic islands in the Pacific Rim - far from anywhere and surrounded by water. There's also a huge magma chamber in the US - under Yellowstone national park, but I'd leave that one alone for now. It's too large, and too close to civilisation to make it a test subject. If something goes wrong - a super-volcano would take out most of North America. And we wouldn't want that, would we! — counterpunch

I can get behind global efforts for clean energy production. Whether your magma chamber idea would be able to produce enough energy is a scientific question, if you're right, that still leaves the political problems of its implementation.

The fundamental nature of the problem is not capitalism. It's our mistaken relationship to science as truth; established when Galileo was tried for heresy, for proving the earth orbits the sun using scientific method. Consequently, we have used the tools, but have not observed the instructions. We continue to act on the basis of ideological conceptions of the world; applying or withholding technology as ideological priorities dictate - and not, as a scientific understanding of reality suggests we should, assuming only we wish to survive. — counterpunch

Do you believe science has an answer to something like: "BLM protesters pulling down the statues they did is praiseworthy because it simultaneously highlights histories of oppression and dismantles symbols of that oppression"?

Any event can be characterized by a statement. Whether or not it ever is, is a separate matter. — Andrew M

I'd be interested in hearing your argument for how you get from:

If a purported event were not representable in language, then we would find ourselves up against the private language argument. We would have no grounds for calling it an event. — Andrew M

To: for every event E possibly there exists a statement S(E) such that E is the truth maker for S(E).

We just need enough energy - and it's there, beneath our feet, a big ball of molten rock 4000 miles deep and 26000 miles around — counterpunch

I see. So you believe drilling to the centre of the planet in lots of places will save us all.

I have a solution, and I know it's right. I can prove it right down to the philosophical roots. I can explain where we've gone wrong and how to put it right in the same terms. I am a philosopher. My core subject is how to save the world. And I know how. — counterpunch

What do you think the problems and the solutions are?

Hard to make contrastive questions out of empty signifiers. "change", "justice", "liberty", "maximal", "limited". "right", "left", "authority", "stasis". What those mean depends on where you would belong on any candidate set of political coordinates.

Event as in space-time region, or event as in abstract proposition about (or property of) such a region? Or something else? Or both? — bongo fury

I don't think the "doctrine of propositional content" as I've been putting it says much at all if all the terms in it aren't fleshed out. And if they are, it seems the doctrine has rather a lot of baggage.

Space time region probably doesn't fit the sense of event or state of affairs as, say, "The USA and Iran backed out of their nuclear deal" doesn't have a space time region associated with it despite it being, allegedly, propositional because it's a statement of fact. Proposition as abstract object doesn't seem to fit either as propositions remain a component part of assertion events.

It seems to me it either says nothing much, says something not particularly evident, or is a coordinating statement of unarticulated metaphysics surrounding it.

For my part, I don't think the doctrine: " "x" and x pick out the same state of affairs" is innocuous at all, since it starts with statements then projects their content into the world; it's like language has cast a shadow onto the world, and the shadow is held up to language, and people are quite pleased with the fit.

I really want to thank you for your help, I can really see the bigger picture here now! — Twinkle221

No worries! I'm glad that my ambiguous "meta language" intension has been cleared up. :smile: This was fun.

I think if we've pinned down the heuristic method, there's still a relevant discussion we could have about how it purports to describe mathematical conduct. Game if you are!

By the time socialists achieve mass working class consciousness, Wall Street will have drowned. — Bitter Crank

:up:

Our old, handy, and familiar categories just don't work very well any more. — Bitter Crank

This is why we must return to Hegel as Lenin did after Oktoberrevolution comrade.

(Marxist yuk yuks)

And yes, the word "metalanguage" felt odd so I'm glad to see we agree upon its use. — Twinkle221

:up:

But that works only when we switch between languages and get rid of the "ordinary language" right? — Twinkle221

I don't know? It seems to me when I read mathematics papers, the majority of the reasoning in them is informal by a logician's standards. Suppressed premises and formally invalid inferences everywhere; but assumed competence and familiarity with the concepts papers over the gaps. I would guess that mathematical reasoning's "mother tongue", so to speak, is natural language, and we enter into formal languages to articulate things with higher precision and replicability. EG, the contrast between the whiteboard doodles two mathematicians working on a paper make when discussing its ideas vs what they actually write in a proof in a paper. Terence Tao has written on a similar theme before; he seems to believe mathematical competence in a domain occurs when one's intuitions in it become formalizable and incorporate previously established theory, not when one can "symbol chase" a proof from definitions to conclusion. To put it in other words, mathematical competence in a domain is marked when one's understanding is not merely formal, but can be formalised.

So I assume, following your answer, that this is why by switching languages we can introduce "real" counterexamples that were not counterexamples before. — Twinkle221

I think one can insist that the L1 definition is correct - barring a monster - and be consistent/not entail contradictions. But if the monster is treated as a refutation, that means rejecting the L1 definition/theorem-statement and massaging the concept/intended interpretation with its related space of hypothetical definitions (which would be L2, L3, L4... if put into words/formalised) until a felicitous matching occurs between them. That matching process has changed "the taxonomic, conceptual..." frameworks of (some of) the involved terms, and those frameworks are expressed in the statement of an L2 definition/theorem-statement in which the monster refutes the L1 statement since the monster now unambiguously counts as an example of the term it targets. EG "Polyhedra are Eulerian" with the intended interpretation of convexity and simplicity vs concave polyhedral monsters.

An intuition pump here might be that "polyhedron" in Proofs and Refutations and "rational number" in my example have ambiguous intensions; we roughly know what is intended by the term, but we don't know all the content required to support our claims using the term. If we imagine we have to treat the intension of those terms (despite their ambiguity) as a constant over our investigation, sticking to one's guns in the face of a heuristic refutation commits no one to a logical error; but it does limit the intension in a way it was not limited before the heuristic refutation - or perhaps the original L1 formulation "said too much", so to speak. As an example of "saying too much", like Euler stating his formula held for polyhedra simpliciter. The monster "becomes" a refutation of the L1 statement by including the monster in the intension of L1's terms, like the picture frame in the intension of polyhedron.

However, if one takes the refutation as an opportunity to include the monster in the intension of the original term, it becomes a refutation, but the intension is also altered tacitly by taking this opportunity.

I'm so sorry, I feel like everything gets mixed up in my head. — Twinkle221

Everything is mixed up in my head too, I'm just hoping writing it out will order things for me.

Especially when you talk about intended interpretation, so this is why there is no contradiction between: (a) all polyhedral are Eulerian, (b) the picture-frame is not Eulerian — Twinkle221

I believe that's so, the kinds of polyhedron intuited in "all polyhedron are Eulerian" are conceptually distinct from the picture frame polyhedron (and the other monsters). What "allows the monsters to work as refutations" is a mismatch between how the concept of polyhedron is articulated verbatim; the content of the definition and theorem statement; and the intended content of the articulation; the restricted, not yet characterised class of polyhedra to which Euler's formula applies.

I think I was infelicitous in using "metalanguage" to talk about the shifts between languages, on reflection. I don't think it's strictly speaking "wrong", but I don't think it portrays the language shifts in the above quote very well. I'm sorry!

So then, by changing language i.e., switching from L1 to L2, we're falling into this meta-language. "If we keep to the tacit semantical rules of our original language our counterexamples are not counterexamples — Twinkle221

I think the key word that marks the presence of heuristic reasoning is "tacit" there. Tacit semantical rules; the latent and hitherto unformalised/unarticulated content of the intended interpretation of the terms.

I guess a clearer example of how I see the language shift thing is:

Imagine I have defined rational numbers as: "a number that results from one number being divided by another". Let's call that an L1 statement. It's L1 because it sets out an initial way of using the term rational number.

Someone like Alpha comes along and writes down "$\sqrt{2}=\frac{\sqrt(2)}{1}$ is a rational number", and it indeed satisfies my definition; since $\sqrt{2}$ is a number, $1$ is a number, and dividing $\sqrt{2}$ by $1$ results in $\sqrt{2}$. So by my definition it is a rational number. I could stick to my guns like Pythagoras allegedly did and claim that sqrt(2) is not a number! A consistent position, but a terrible one nevertheless. I could alternatively accept the counterexample... But that would mean adopting new standards of interpretation and ways of talking about the concept of fraction; since it's no longer right to construe it as "a number that results from one number being divided by another". To make that shift, I've got to change how I write and think about fractions, and set out new rules to characterise them. So I'll accept Alpha's concept stretch refutation and institute a new way, an L2.

I come back to Alpha with: "A rational number is the result of dividing one integer by another integer".

To explain why I made that change, I need to explain how my previous articulation of rational numbers was wrong; it just emphasised that you can write one number on top of the other, and Alpha did just that with a monster on the top. So there was something else other than "one number on top of the other" that I needed to include, to a first approximation I'd guess instead of "dividing one number by another number", I'd restrict it to "divide one integer by another integer", that should stop Alpha from coming back with any counterexamples like that - square roots etc are not integers. Perhaps this explanation is in some other language, L3, in which I compare L2 to L1... Comparisons between fraction concepts do not seem to be part of the terms I came back to Alpha with.



Alpha might hate me for monster barring because $\sqrt{2}$ is a perfectly good number and I've barred that by putting in "integers" explicitly into the definition to block it, but I could've also done "A rational number is a number that results from one number being divided by another except $\sqrt{2}$", which doesn't refine the understanding of rational numbers at all, since it doesn't explain why the heuristic refutation forced me to refine my definition - ie, it doesn't spell out how the heuristic refutation clashed with the intended interpretation of the terms. Putting "integer" in there blocks some of the right monsters, so to speak.

And I guess we could add that concept formation cannot be separated from "definition formation" which relates to a more heuristic study of knowledge. — Twinkle221

That's my reading too. Wrestling with definitions is how you pin down the intended interpretation in mathematical language, and that "pinning down" is an articulation of the content of the mathematical structure under study.

In terms of the languages, I'd guess that the theorem statement is in L1.

X is a sandwich if X is a food item consisting a food item (called "the filling") bounded between two pieces of bread.

It doesn't contain a contradiction by itself, it just perhaps doesn't fit what is the intended interpretation of a sandwich. When someone provides an example of a food item which fits the definition but not its intended interpretation, they pass into a meta language L2 to present the discrepancies between the intended interpretation of the terms in the theorem and the counterexamples. And when discussing them, they're already in a meta-meta language L3 in which the intended intepretations of L2 statements and their relationships to L1 terms (like in the theorem statement) are weighed.

I think whenever you start talking about the meaning of the terms in a language, you pass into a meta-language about it.

Compare this statement of semantic completeness of a formal system S.

If X is semantically derivable in S then X is syntactically derivable in S.

You have the level of formulae like X, then you have the level of proofs on formulae and interpretations of formula which syntactic and semantic derivability is talking about, then you have the embedding in another (somewhat informal) metalanguage that spells out the connection (if-then, in which language is that implication relation defined?) between the semantic derivability and syntactic derivability.

I finished rereading this book last week, I can tell you my take on heuristic counterexamples.

Imagine you've got this "mathematical theorem" which has been conjectured but not yet given a formal proof...

(1) X is a sandwich if X is a food item consisting a food item (called "the filling") bounded between two pieces of bread.

A so called logical counterexample to it would be something that satisfies all the premises with their intended interpretation but not the conclusion with its intended interpretation. In this case, it would be a food item that has a filling bounded between two slices of bread which is not a sandwich... Perhaps burgers count for that, since they're not clearly sandwiches, but do have a filling (in a sense) and the filling is between two pieces of bread. So burgers perhaps would be a logical counterexample to the characterisation of sandwiches as "sandwiches are all and only those food items that have a filling bounded between two pieces of bread".

Someone like Alpha in the book would present this monster to challenge it:



Because it is a foodstuff whose position is bounded between two pieces of bread, and is thus a sandwich by the definition. A lot of concept stretching has gone into that example; is a banana with the skin on a food item? Does a banana count as a filling if it has the skin on? Do whole pieces of bread separated in space with a banana placed inbetween them count as a sandwich?

The first example of the burger grants the intended interpretation of the terms in (1); the bits of bread bun in the burger resemble slices etc. the second example intentionally stretches the intended interpretation while staying within the letter of the theorem statement. The first looks logical since it grants the intended interpretation and simply blocks the inference; it ain't a sandwich, it's a burger, but it is a food item bounded between two pieces of bread. The second looks more heuristic since it seems to stretch the concepts of the terms involved while staying within the letter of how they were used in the theorem.

Both the burger "logical" counterexample and the monster "heuristic" one actually do the same thing (for Lakatos, I think), they reveal that the intended interpretation had hidden content/assumptions which were not expressed in the verbatim statement of the theorem. The refutations could lead to a revised conjecture:

(2) X is a sandwich if and only if X is a foodstuff, it has a foodstuff Y which is prepared and edible called the filling, the filling is located snugly between two slices of bread, and the two slices of bread enclose the filling.

Which might inspire more counterexamples - Alpha might put Y="a small piece of bread", and now it's a bread sandwich...

No. An event does need to be representable in language, in principle (i.e., such that language users could potentially make a statement S(E)). But it need not actually be represented by someone in practice, now or ever. — Andrew M

So your claim's more like:

(DPC) For every event E possibly there exists a statement S( E ) such that E is the truth maker for S( E ).

?

When will you be able to get it? — frank

Not expecting to any time soon. September last I heard.

Nice. Had a heads up from a friend about the Oxford vaccine, it seems if your body already recognises corona your immune system goes nuts for a day or two and it's horrible. He was mostly fine after the first day he was ill.

Do you believe that it's a property of the world that whatever happens in it can potentially be stated? — fdrake

Yes. — Andrew M

:up:

Let's put it in more formal terms, I'll label the formalisation (DPC) for "doctrine of propositional content".

(DPC) For every (*) event E there exists (**) a statement S( E ) such that E is the truth maker for S( E )

Do you agree with that formulation?

The implicit domain of quantification of the existential quantifier ( ** ) hides rather a lot of the issue, I think. Here are a few thoughts about assigning a domain to each quantifier.

(1) The class of events ( * ) universally quantified over is all events at all times. The second quantifier (**) is given the same domain.

This raises a question of time.

Say if you truncated the domain to events which occurred 10 billion years ago, at that time there would be events but no statements, so the existential quantifier is false since there do not exist statements at that time, so the whole thing would be false because there would be events without statements - and thus events without corresponding statements.

To restore the truth of the conjecture, those troublesome events which occurred 10 billion years ago can find their corresponding statements now. That seems fine, as the domain is posited to be all events and all times. However, DPC then turns on this hypothetical connection between events which occurred 10 billion years ago and statements now for all events 10 billion years ago. If there did not exist an S(E) at the time of E, S(E) must have occurred later. In turn, that raises the questions, why would DPC be a property of the world if it only applies after the advent of declarative sentences? If it's part of metaphysics, it's not part of the metaphysics of nature; it did fine in the truncated domain.

(2) One way of sidelining this issue would be to claim that the universal quantifier is over all events and all times, but the existential quantifier ranges only over the class of all statements. That gives us a few sub cases:
(2a) statements are events, effectively this is a collapse back into case (1)
(2b) some statements are events, some statements are not events - this is very wooly, under what conditions is a statement an event and under what conditions is it not? I'll not consider it further unless I have to.
(2c) No statements are events, all events are not statements.

Left with (2c), this commits someone to the doctrine that statements are not events; and if they are not events they could not be asserted (unless assertions are not events!). That raises questions regarding how statements work if they need not be the events of their assertion. An ontology of statements as distinct from speech acts. In this case, DPC seems to turn on a construal of statements as a kind of abstraction which is distinct from the speech acts of their assertion.