Philosophy of Mathematics — Numbers, Sets & Structures

0:00 Um, he complained about treated collections of mathematical entities as completed totalities. The classical mathematician, quote, introduces various concepts entirely meaningless to the intuition. Such as, for instance, the set whose elements are the points of space, the set whose elements are the continuous functions of a variable, the set whose elements are the discontinuous functions of a variable, right, end quote. So as Brauer sees things, the divergence between classical and intuitive mathematics also has an epistemic dimension. So far, we focus on these sort of, or mentioned quotes, the metaphysical differences between them. But also, there's an epistemic side of it. So this is the quote that's on the slide. the point of view that there are no non-experienced truths has found acceptance with regard to mathematics much later than with regard to practical life in the science. I'm kind of curious how he's writing, like he's already won all these battles, hasn't he? He always does write that. Yeah, right, it's found acceptance, like everyone agrees with him. Anyway, mathematics rigorously treated from this point of view, that is, the point of view that there are no non-experienced truths, there are no unlawful truths, including deducing theorems exclusively by means of an introspective construction is called intuitionistic mathematics classical mathematicians believe in the existence of unknown truths and in particular apply the principle of excluded third expressing that every mathematical assertion either is a truth or cannot be a truth alright that's the end of that quote now Brouwer's student R.N. Hayten reiterated the focus on mathematical construction again here's a quote from We do not attribute any existence independent of our thought, i.e. a transcendental existence, to mathematical objects. Mathematical objects are by their very nature dependent on human thought, in this situation. Their existence is guaranteed only insofar as they can be determined by thought. They have properties only insofar as these can be discerned in them by thought. Faith in the transcendental existence must be rejected as a means of mathematical proof. This is the reason for doubting the law of excluded middle. Now with this teacher, Hayten argued that classical mathematics relies on a metaphysical principle. Again, that was a word that Hayten used. That it relies on a metaphysical principle that the truths of mathematics are rejected,

2:30 or that they're truly independent of the mathematician's mind or something. And he said the only way to avoid what he calls a maze of metaphysical difficulties them from mathematics, that is, to banish these metaphysical principles from mathematics. Again, so wait for the next slide. Alright, if to exist does not mean to be constructed, it must have some metaphysical meaning. It cannot be the task in mathematics to investigate this meaning or to decide whether it's tenable or not. the mathematician privately admitting any metaphysical meaning he likes, but Brouwer's program intended to study mathematics as something simpler, more immediate than metaphysics. And the study of mental mathematical constructions, too, must be synonymous with to be constructed. All right, now moving up to the 20th century, well, I guess we're in the 20th century, moving up to the second half of the 20th century. Some decades later, I guess we already were there, too. Moving up to closer now, Michael Domit shifted the debate from the arena of metaphysics to semantics. So far, except for that quote from Brouwer, we seem to have a metaphysical dispute. But Domit recasts it as a semantic one, into philosophy of language generally. Starting with his 1973 paper on the philosophical basis of linguistic logic, he argued that any considerations concerning logical principles must ultimately turn on questions of meaning. He thus adopted a now-wide-held view that the rules for drawing inferences from a set of premises flow from the meaning of some of the terms in the premises, the so-called logical terminology. Since language is a public medium, the meanings of the terms in a language are determined by how the terms are correctly used in discourse. Again, so here's a quote from Donnett from that same article. This may be the one that's up here. The meaning of a mathematical statement determines and is exhaustively determined by its use. The meaning of such a statement cannot be or cannot contain as an ingredient anything which is not manifest in the use of the use, lying solely in the mind of the individual who apprehends that meaning. If two individuals agree completely about the use to be made of a statement, then they agree about its meaning. the reason is that the meaning of a statement consists solely in its role as an instrument of communication between individuals

5:00 an individual cannot communicate what he cannot be observed to communicate if an individual associated with a mathematical symbol or formula some mental content where the association did not lie in the use he made of the symbol or formula then he could not convey that content by means of the symbol or formula or his audience would be unaware of the association and would have no means of becoming aware again, this is a very much discussed passage This common sense view of language, at least so far, when it seems to be common sense, view of language supports stuff its manifestation requirement, a thesis that anyone who understands the meaning of an expression must be able to demonstrate that understanding through her behavior, through her use of the expression. Here's the quote itself. Nope, this isn't it either. All right. Here's another quote that's on the handout. There must be an observable difference between the behavior or capacities of someone who is said to have knowledge of the meaning of an expression and someone who's said no Hence it follows that a grasp of the meaning of a mathematical statement must in general consist of the capacity to use that statement in a certain way or to respond in a certain way it was used by others. Now, again, Dalmat argues that these considerations have ramifications for the proper meaning of the logical terminology. In the prevailing Tarskian semantics, the truth conditions of a complex formula are defined in terms of the truth conditions of its subformulas. Typically, truth conditions can hold or not, independent of our abilities to know that they do. This is a source of bivalence and exclusion melt. Dummett argues that if the language is undecidable, a semantics like this, that is, this Tarski and bivalent one, violates the manifestation requirement. On a classical bivalent interpretation of the mathematical theory, finally the quote on this slide, quote, the central notion is that of truth. That is, on a classical bivalent interpretation, notion of the mathematical theory is that of truth. A grasp of the meaning of a sentence consists in a knowledge of what it is for that sentence to be true. Since in general, the sentences of the language will not be ones whose truth value we are capable of effectively deciding, the condition for the truth of such a sentence will be one which we are not in general capable of recognizing as attaining whenever it attains, or of getting ourselves into the position in which we can't so recognize it. All right, that's the end of the book. Dummick claims that verifiability or assertibility should, so Dummick claims then that verifiability or assertibility should replace truth as the main constituent of a compositional semantics.

7:30 In mathematics, verification is proof. So Dummick's proposal thus invokes the central theme of faking semantics for intuition-specific logic. Instead of providing true conditions for each formula, we supply proof conditions. In rough terms, oh, here's some clauses again the next slide. Here are some clauses for Hagen semantics, and again, this is very rough, and I know you can put some of these if you want. But here's a more or less typical, at least, first approximation of Hagen semantics. So a proof of a sentence of a form phi or psi consists of either a proof of phi or a proof of psi. A proof of a sentence of a form if phi, then psi, consists of a method for transforming any proof of phi into a proof of psi. A proof of a sentence in the form not phi, you can typically, intelligentists, think of not phi as phi arrow absurdity. So it would consist of a procedure for transforming any proof of phi into a proof of absurdity. But of course, there are any proofs of absurdity. So in other words, a proof of not phi is a proof that there can be no proof of phi. A proof of a sentence in the form for every x phi consists of a procedure, again, produces a proof of the corresponding sentence phi n. And a proof of a sentence in the forming cis-sex phi produces the construction of an item n and a proof of the corresponding phi n. So proof conditions, which is how we're going to stipulate or give the meanings of propositions, are done in terms of prior proofs and procedures. Now, notice that in this framework, presumably, one cannot have a canonical proof of a disjunction. for now that's the relative one unless one has a proof of one of the disjumps so one cannot have a canonical proof of an instance of excluded middle phi or not phi unless one has either proof of phi or proof that there can be no proof of phi so if the language is undecidable then many instances of LEM are not justified by hating semantics and if this semantics sufficiently reflects the many mitological constants then typical instances of LEM cannot be justified on the basis of meaning alone So in general, excluded middle is not analytic, and so is not logically true. As Dometh puts it, a major presupposition of classical mathematics is that there are or may be truths that cannot be known. A truth-valued bivalent semantics suggests that truth is one thing and no ability another.

10:00 Again, this is sort of from the realist perspective. Truth is one thing, no ability, and another. Domet's view, sometimes called global semantic anti-realism, opposes this. The anti-realist holds that, at least in principle, all truths are noble. The possibility of an unknowable truth is ruled out a priority. When the discourse is undecidable, intuition is the logic reflects this anti-realism. As we saw, the epistemic dimension of Brouwer's thought points towards semantic anti-realism for mathematics, as well as what he calls for practical life in science. Remember, for Brouwer, there are no non-experienced truths. So that seems to be roughly the same thing as the Median anti-realism. Now, it's common to point out that it is hard to adjudicate the clash between classical mathematics and the dumbest semantic anti-realism. Since the parties to the dispute did not mean the same thing by the logical terminology, right? I mean, the dispute is supposed to turn on meaning, right? And the classical mathematician doesn't mean the same thing as the intuitionist, so how can they possibly adjudicate their dispute? The classical mathematician might cheerfully concede that, of course, excluded middle is not logically true on the intuitionistic, that is, the 18-Dumick understanding of the connectives, but excluded middle is an obvious analytic, logical, whatever you want to call it, the truth on the classical understanding of the connectives. The instance, phi or not phi, just says that phi is either true or false. Not that phi is either provable or refutable. And what could be more obvious than that? So we thus broach what is sometimes called the problem of shared content. Which is that we need to have shared content in order to actually have a meaningful debate. Prima facia, the intuitionists and the classicists talk past one another since they do not mean the same things by their logical terms. In one place, I'll leave for another slide, Dometh puts the problem this way, but he maintains that the classical terms have no legitimate meaning. I don't like particularly like the example that he uses, but I think it'll do to illustrate the point. You can see the bottom, we're almost ready for the fractal cycle part. Let's see, so here's Dometh. The intuition is told that certain methods of reasoning employed by classical mathematicians and proving theorems are invalid. The premises do not justify the conclusion.

12:30 The immediate effect of a challenge to fundamental custom modes of reasoning is perplexity. Again, you often see this, right, from classical mathematicians when you first hear about this. On what basis can we argue the matter if we are not in agreement about what constitutes a valid argument? The affront to which this challenge gives rise is quickly allayed by a result to take no notice. That's another common response, by the way. That is, to just not bother listening. The challenger must mean something different by the logical constants, so it's not really challenging the laws that we have always accepted and may therefore continue to accept. This attempt to brush the challenger aside works no better when the issue concerns logic than in any other case. So here's the example. Again, I'm not sure that the example works, but I think we get the idea of what that's up to. Perhaps a polytheist cannot mean the same by God as a monotheist, disagreement between them all the same. Each denies that the other has hold of a coherent meaning, and that is just a charge made by the intuitionist against the classical mathematician. I get that. It's a strong charge, right? The classical logical terms have no coherent meaning. He, who's the he? The intuitionist, acknowledged that he attaches meanings to mathematical terms different from those that the classical mathematician describes to them. But he, the intuitions begin, maintains that the classical meanings are incoherent, and arise at a misconception on the part of the classical mathematician about how a mathematical language functions. Thus, the answer to the question, how it is possible to call a basic logical law in doubt, is that, underlying the disagreement about logic, there is a yet more fundamental disagreement about the correct model of meaning, that is, about what we should regard as constituting an understanding of the statement. All right, so that's the end of the first section. The second section is called, Can We Still Be Friends? Now, no one doubts that classical mathematics has been enormously successful, at least in its own terms, and that classical mathematics is deeply entrenched in the scientific web of belief. I mean, I know that there's a very serious research program just to see how much classical mathematics you need, and my chair here is deeply involved in it, but, you know, at least for now, it seems to be pretty deeply entrenched in the scientific web of belief. Now, whether it's necessarily so or not, it's But mathematicians are loath to give up successful methods, even in light of what looks like cogent philosophical criticism. Some philosophers follow suit. That is, being loathed to give up something that's so successful.

15:00 When faced with a conflict between classical mathematics and intuitionistic philosophy, they reject the philosophy, perhaps out of hand. There must be something wrong with intuitionistic arguments, either Brouwer's metaphysics and epistemology or Domington's views on language, if they lead us out of our classical paradox. In a similar context, and with characteristic wit, David Lewis wrote, I think you don't like this, he's absolutely, well, he's often quite fine, but I think he'll like this one. I'd like to think how presumptuous it would be to reject mathematics for philosophical reasons. How would you like to go and tell the mathematicians that they must change their ways? Will you tell them with a straight face to follow a philosophical argument wherever it leads? If they challenge your credentials, will you boast the philosophy's other great discoveries? That motion is impossible. That it's unthinkable that anything exists outside the mind. That time is unreal. And that no theory has ever been made at all probable by evidence. That it is a wide open scientific question whether anyone has ever believed anything. Not me. Well, I guess not me either, but... Brouwer, again, this has been a later paper, a 1948 paper, Brouwer conceded that classical mathematics may be what he calls appropriate for science, right? So Brouwer made a rather interesting concession. He said that classical analysis may be appropriate for science. Nevertheless, he maintained that classical analysis has less mathematical truth than intuition of the analysis, since the former runs against the mind-dependent nature of mathematical construction. Brouwer thus proposed a bold divorce between mathematics and the empirical sciences. So the scientists have our needs, but there's the true mathematics, and they have to come first. Other thinkers may be more reluctant to jettison classical mathematics. After all, our philosophical arguments are rarely as compelling as mathematical demonstrations. But of course the issue here concerns just what is or ought to be a compelling mathematical demonstration. Hinting's early papers, or the early paper I was referring to here, it was a 1931 paper. writing, echoes Brauer's claim that classical mathematics is flawed and should be replaced by intuitionism. So again, one sentence there. Intuitionism is the only possible way to construct mathematics. That's what Brauer said early on. However, his 1956 book is more eclectic. Arguing only that intuitionistic mathematics deserves a place alongside classical mathematics. That was a phrase used, at least in the English version. I think the book was written.

17:30 That mathematics deserves a place, sorry, intuitionistic mathematics deserves a place along with classical mathematics. The later Hating did not claim a monopoly on mathematics and would rest content if the classical mathematician admits the good right of the intuitionist conception. Nevertheless, Hating maintained that the intuitionist is doing things cleanly and honestly, while the classical mathematician relies on a dubious metaphysical principle. For what it is worth, though, by the time of the 1956 book, it was pretty apparent that intuitionism and the colossal war for the hearts and minds of mathematicians, and you have sort of wanted to show, well, look, the intuition's de-mathematics is still a legitimate study, and, you know, let's sort of keep it on the table. Dometh joins Brouwer, now, Dometh joined, for his part, Dometh now joins Brouwer in the early aging and having no trouble with an eclectic orientation. He's not interested in softening the disagreements, nor in making some space for classical mathematics. Dometh, in 1973, explicitly says that he's exploring the thesis that, quote, classical mathematics employs forms of reasoning in the way of construing mathematical statements. End quote. So Dometheus tells mathematicians that they must change their ways, to use Lewis's phrase. In a sense, Dometheus further than Brouwer and Aintin, and even the early Aintin. The two of them just accuse the classical mathematician of harboring a false, dubious, or unwarranted metaphysical view. The Dometheus accuses the classical mathematician of incoherence a serious charge indeed all right now perhaps there is a less confrontational stance within the general of the median framework one one who buys the thrust of damit's argument might claim that the basic principles of intuitionistic logic enjoy a certain type of justification they are true in virtue of the meaning of the logical terminology that is the intuitionistic theses are when the terminology is construed in something like the done of 18 man most instances of excluded middle do not enjoy this level of justification, and neither do those parts of classical mathematics that rely on excluded middle. This is not yet to say that classical mathematics is a coherent, only to note that parts of it have less justification than other parts. The classical parts, again, are justified not in the same way as the intuitions in parts. If the classical mathematician accepts this assessment, she would probably go on to claim that classical mathematics does not stay in need of this extra justification. She might patiently point out that mathematics has its own rigorous standards for correct and incorrect proof,

20:00 and that these have stood the test of time, perhaps thanking the philosopher for his interest just the same. It comes down to personal standards for justification. But notice that this uncomfortable piece, or mutual tolerance, requires us to somehow finesse the problem of shared content. And now we're going to turn to the Tenet's work. So section three, attempted reconciliation. Neil Tennant, a prominent Dometian, proposes a way for classical mathematics to enjoy a level of support and justification, even from the perspective of the semantic anti-realist. Tennant notes that the, quote, the realist does not wish to be methodologically deprived, end quote. The realist is, quote, unwilling to give up the strictly classical inferences of classical logic, end quote. If Tennant's realist is a classical mathematician, this is quite correct. Apparently, classical mathematicians remain unconvinced that their mathematics is broken, and to use the old expression, if it ain't broke, don't fix it. Tennant resolves the problem of shared content in favor of the Domenian intuitions. From the start, he insists, quote, with Domet on the manifestation requirement in the theory of me, end quote. And he advocates a, quote, far-reaching logical reform. Actually, Domet's far-reaching logical reform is more far-reaching than Domet. He thinks we need intuition, psychologically, and relevant logic. mixed them together. That's just one true logic. But we're not going to focus on the relevant parts. Sorry, we will focus on the relevant parts. We won't focus on the relevant logic parts. For Tennant, the realist is not entitled to strictly classical inferences. In particular, he agrees with Dummett that there is no coherent meaning for the logical terminology that makes excluded in their own analytic. However, Tennant then tries to spell out how the realist can be allowed to have his cake, but also be asked to improve his manners when eating it. That is, this is on slide four. I mean, this is on this slide. Tenant, thank you. Yeah. Tenant is out to present an anti-realist's take on classical mathematics. So, quote, the reforming anti-realist is not so much intent on depriving the realist of his classical tools, but rather on having the realist acknowledge the true nature of his use of them, or of his appeal to them, end quote. Now, as a Demetian intuitionist, Tannen holds that most instances of excluded middle are not true in virtue of the correct meanings of the logical terms. The logical terms here being disjunction and negation.

22:30 So, excluded middle is not analytic. It's not true in virtue of means. Thus, most instances of excluded middle are synthetic, if true. If the classical mathematician still claims to know each instance of excluded middle a priori, then she must regard most instances as synthetic a priori, thus reviving the status that condesigned to most mathematics, that is synthetic a priori. Hemant's title, then, of the paper that we'll be talking about today, The Law of Excluded Middle of Synthetic a Priori is Valid, is, now this is his words, quote, in a certain sense, a vicarious claim put forward by the perspicuous anti-realist on behalf of the realist, end quote. So, in fact, he's telling us, realists, or us classical mathematicians, what we need to do in order to keep classical mathematics on the table. The issue here is whether the classical mathematics can accept this proposal, adopting the Hainting-Dumick understanding of the logical terminology, and then accepting excluded middle as a necessary but synthetic and so non-logical truth, perhaps still knowable of priority. So Tennant's proposal is quite similar, actually, to Haydn's claim that excluded middle embodies a metaphysical principle. Remember, we heard that earlier. Haydn says that there's this metaphysical principle that underlies classical mathematics, which is not part of mathematics itself. For Tennant himself, this optional metaphysics is perhaps dubious, but the classical mathematician is free to hold it. So again, this is on the next slide. By the way, there about how far we're getting. At least you know when this agony will be over. All right. All right, so let's see. So this is a quote from, I'm not going to read, I'll read part of this. The holding true as a matter of necessity of every instance of excluded middle, or the holding true of any such instance, P or not P, absent any proof or refutation of P, expresses an essentially metaphysical belief. The belief is that the world is determinate in every expressible regard, or in this case, with regard to p. Such a belief is synthetic, since its content cannot be known to be true simply on the basis of the meanings of logical expressions, or and not. I'll skip the other one. Tennant has a small favor to ask of a classical mathematician. then, the classical mathematician. Once she acknowledges the synthetic metaphysical nature of excluded middle, then she should note each use of it

25:00 improves, so that, quote, we can always be clear as to how and when the justification of a knowledge claim involves recourse from the sexually metaphysical conception of reality. All right, so he asks the classical mathematician just to flag every use of excluded middle because that's the place where this metaphysical principle is being invoked. All right, so whenever the classical mathematician or not be, she, quote, must be prepared to acknowledge, theorize, resuppose or give expression to the metaphysical view that reality is determinate in respect to people, end quote. This way, the realist metaphysical outlook can be logically corralled as technically. If the classical mathematician makes this explicit acknowledgement, then the mediating anti-realist would no longer wish to deprive her of anything. So, now we can all get along famously. Or so it seems. Tenet's picture, then, is that there is no problem of shared content. Classical mathematics employs the same language as intuition is deep mathematics. In particular, the logical connectives have the same meaning in both discourses, and knowledge of this meaning does not suffice to justify instances of excluded middle. Contemporary philosophers who maintain an analytic-synthetic distinction thereby also invoke a distinction between sentences true in virtue of meaning since the classical mathematician's insistence on excluded middle does not have a source in meaning, then it must be the world that is responsible for excluded middle this commitment on the part of the classical mathematician is expressed in the metaphysical principle that the world is determined language does not underwrite the determinacy, the world does as Tenet puts it, the classical mathematician is making a synthetic claim about reality itself appropriately aided thereto by the anti-realistically illicit meanings logical operators. Okay, let's skip a bit. On Tenech's view, then, there is little or no revisionism required of the classical mathematician. All she is asked to do is to practice a little logical hygiene and acknowledge the metaphysical principle of determinacy. The intuition is demurs from this metaphysical principle and also from most instances of excluded middle. Recall that some parts of intuition is deep mathematics, such as Brouwer's theory of free choice sequences, or actually Brouwer's theory of real analysis, that all real-valued functions are continuous is actually inconsistent with classical mathematics. So some of Brouwer's actual mathematics is actually inconsistent with classical mathematics.

27:30 Call the theory of free-choice sequences and the like Brouwerian mathematics. The classical mathematician, of course, must reject Brouwerian mathematics, but since our intuitionist does not accept the excluded middle, he is free to accept it. Perhaps Brouwerian mathematics is itself based on metaphysical principles concerning the underlying nature of the real numbers. The Brouwerian may claim that these metaphysical principles are synthetic a priori, whatever it is that this notion is best. Apparently, these principles conflict with the classical mathematician's equally synthetic principle of determinacy. And so this battle will take place on metaphysical plans. When it comes to metaphysical principles, perhaps, you pay your money and you make your choice. or you can play it safe and stick to those parts of intuitionistic mathematics that are classically correct following the issue adopting neither of the metaphysical principles anyway, section 4 called It's a Rotten Deal on tenets view then the classical mathematician and the intuitionist speak the same language and so they can follow each other's discourse and get along provided that each of them explicitly acknowledges any metaphysical principles in play in what follows is not as simple as this. Once the classical mathematician accepts the basic Dunnett-Tenet framework, she is saddled with a very implausible view, even if this view should turn out to be consistent. So the classical mathematician should decline Tenet's invitation with thanks, and indeed the whole domain framework for me. See, I'm almost ready for another slide, I think. One problem is that the arguments that underlie Dunnett-Tenet's revisionism point toward a very general and quite global anti-realism. The meanings assigned to the logical terms, the only meanings that these terms can have, are part and parcel of this anti-realism and cannot be jettisoned from it. The slogan of anti-realism is that all truths are knowable, truth is epistemically constrained, to use Crispin's phrase. Let's see. Slide six. Oh, we're getting there. All right, so once the classical mathematician accepts the domain framework, which flows from the arguments establishing the meaning of opological terminology, she must claim to know, somehow, each instance of excluded middle, I mean, after all, the claim isn't according to Tennant's framework, it's synthetic a priori, so it's noble.

30:00 She must claim to have an, quote, effectively checkable construction establishing each instance of excluded middle as true. does the mere adoption of the metaphysical principle that underlies excluded middle provide such a warrant? I mean, there is some tension, actually, right from the start. Recall how a tenant formulates the metaphysical principle that underlies excluded middle. This is back to that slide five. It's the one I didn't read, I think. According to the realist principle of determinacy, the truth value declared a sentence is determined by reality in advance of our investigation. So this is the metaphysical principle that supposedly underlies exclusive health. And the truth value could attach to a statement quite independently of us and our beliefs, and also independently of our available means for coming to know what is the case. Now this seems to be an outright contradiction to Tenet's statement of anti-realism. Truth is knowable and consists in the existence of an effectively checkable construction or warrant establishing the sentence in question as true. One who accepts Dumit's arguments concerning the meanings of the logical terminology should accept the anti-realism that flows from those arguments, and so such a person should completely reject the metaphysical principle in question that reality is determined, independent of I see only one resolution to this, if you want to maintain that this thing is consistent. Dumit himself concedes that if a discourse is decidable, then excluded bill holds for means in a minute. Tenet's summary of the debate conclusion is a converse to this, quote, one cannot hold the bivalence in manifestationism for any undecidable discourse, end quote. Right now, a discourse is decidable if there is a procedure for determining of any given sentence fine in the discourse whether phi is true or false. So if a discourse is decidable, then for any sentence fine in the discourse, either phi is provable or phi is refutable. Presumably, then, according to everyone, we can hold to bivalence and manifestationism only for decidable discourses. Sorry, according to everyone, we can hold to bivalence and manifestation for decidable discourses. According to the Dometians, only for decidable discourses. Now, the principles of semantic anti-realism yield a tight argument for this conclusion, independent of many subtle issues of the meaning of the logical terms. That is the principle that all truths are knowable. Hating semantics...

32:30 Independent on any subtle issues that turn the meaning of the terms, taking semantics, or metaphysical principles of determinacy. With intuitionistic logic, we can show that the citability of every proposition follows from excluded middle and the anti-realist principle that all truths are noble. Again, it's a simple argument. So, excluded middle, phi or not phi. and then if phi so assume phi and we get phi is knowable from anti-realism and then we get phi is knowable or refutable and then we assume not phi and then you know get not phi is knowable and so either phi is knowable or refutable not phi is knowable is the same as phi is refutable alright so we get that middle and anti-realism alone, we get that the discourse has to be decidable in the prevailing sense. So anti-realism and excluded middle entail that either phi is provable or phi is refutable. As noted above, we'll go back to slide six now. Dumont's arguments for manifestation, harmony, and separability suggest that the logical connectives be interpreted along the lines of making semantics. And this sharpens the conclusion of the last two paragraphs. How hard does this conclusion consider? If you actually look to see what the connectives mean. I don't know why I want this. But anyway, don't want to dig out the other one where Haydn's Semantics is. According to Haydn's Semantics, the assertibility conditions of a sentence of the form phi are not phi. So, according to Haydn's Semantics, the assertibility conditions for this will consist of a proof of phi or a refutation of phi. So if the classical mathematician wants to assert this based on a metaphysical principle, then what he's asserting is that he has a proof of phi, or knows how to find one, or he has a refutation of phi. Alright, so if someone asserts an instance of excluded middle, then she claims to know that for the embedded sentence phi is either provable or refutable. Alright, so we're led to the following conclusion. Since Tennant's classical mathematician accepts the Medellin framework and the insuling into realism, then it's classical mathematician, right? Then she must hold that classical mathematics is decidable. Every sentence is not approvable or refutable.

35:00 From the perspective of global into realism, the metaphysical principle that underlies classical mathematics is not just a view about the way the world is. The metaphysical principle tells us that the world is cooperative. So remember, the way Tennant puts it, the principle is that the world is determinate. But given anti-realism, it would fall from that the world as cooperative, eventually revealing all of its secrets, deciding all of the sentences of mathematics one way or the other. Now, in an earlier paper in a different context, I define optimism to be the view that, in principle, every unindiculous sentence of mathematics is either provable or refutable. In the context of semantic anti-realism, excluded middle amounts to or entails optimism. Actually, when Jeffrey Hellman made this paper, he called it the hubris, the statement that we can know everything. If one takes historical evidence to be relevant to the issue of optimism, it goes both ways. The optimist can point out examples of problems like the four-color result in Fermat's last theorem that remained open for a long time, but now have a generally accepted resolution. Against this, one can point out examples of very old problems like the Goldbach conjecture that remain open to this day. As great a mind as Gertle once endorsed optimism, human reason is not utterly irrational by asking questions it cannot answer, while asserting emphatically the only reason to answer them. Those parts of mathematics which have been systematically and completely developed show an amazing degree of beauty and perfection. In those fields, by entirely unsuspected laws and procedures, means are provided not only for solving all relevant problems, but also solving them in a most beautiful and perfectly feasible manner. This fact seems to justify what may be called rationalistic optimism. Sometimes this view is called redilting optimism. I think 10 is big about that, probably from that paper of mine. The opening of the Hilbert's Celebrated Mathematical Problems lecture in 1900 is also an enthusiastic course in the optimism, but I won't bother going over that quote. The anti-realist rules out unknowable truths on a priori conceptual grounds concerning the nature of truth. For the anti-realist, truth itself has an epistemic component. If there can be no proof, there can be no truth either. So anti-realism has no consequences

37:30 is concerning the powers or limitations of the human mind, interrealism by itself doesn't. But once she adopts excluded middle, the anti-realist thereby adopts a thesis that the human mind is capable of deciding every non-ambiguous mathematical proposition, and this is a substantial thesis of our species. One might very well wonder whether our minds are that powerful. In a discussion of Gödel's optimism, Boulos, it's actually an introductory note to the paper in the collective volumes, wonders, quote, why should there not be mathematical truths that human minds can comprehend. And again, why not a D? Now, we know from Gödel's complete incompleteness theorem that for any sound-effective formal system S of arithmetic, there is a sentence phi that is not a theorem of S. There is a true sentence phi that is not a theorem of S. According to optimism this sentence S is knowable. That is, the formal system is knowable. So it follows from optimism that, oh sorry, that shouldn't be, I guess. According to optimism, phi is noble, right, because it's true. Phi would be the sort of standard Gerdel sentence. So it follows from optimism that there is no effective formal system that captures all and only the noble sentences of any mathematical theory as rich as arithmetic. So optimism is inconsistent with the mechanistic thesis that the noble arithmetic propositions are recursively innumerable. I mean, some mechanists might believe this. Gerdel himself came to a similar conclusion, and again, in a very, in a typically careful way. is this quote is the one that's up here. The following disjunctive conclusion is inevitable. Either mathematics is incompletable in the sense that its evident axioms can never be comprised in a finite rule, that is to say, the human mind, even within the role of pure mathematics, infinitely surpasses the powers of any finite machine, or else there exists absolutely unsolvable problems that the type specified. Again, you can see from his optimism that he would go with the first disjunct rather than the second. Boulos will probably go with both. It is this mathematically, quote, get back to Gödel, it is this mathematically established fact which seems to me of great philosophical interest. All right, end quote. As always, Gödel's conclusion is careful. Either optimism is false, and there are unknowable propositions of arithmetic, or else mechanism is false. Or both, of course. Of course, our topic here is excluded middle and anti-realism, not mechanism. Gödel's conclusion is quite consonant unanimous claim that mathematical provability is inherently informal and is not completely codified in any effective deductive system. When the anti-realist declares that all truths

40:00 are knowable, he means knowable in principle, with no effective limits placed on how sentences can become known. Similarly, when Tennant's classical mathematician, our optimist, declares that all sentences are decidable, she means decidable in principle, not effectively decidable. Notice that the mechanistic thesis that the noble arithmetic truths are recursively enumerable is already inconsistent with the anti-realist thesis that all truths are noble. We've just been over that. Under anti-realism, the mechanistic thesis is that arithmetic truths are recursively enumerable, and this contradicts Tarski's theorem on the undefinability of truth, noting that Tarski's theorem is intuitively acceptable. Even so, in the context of anti-realism and Heike's semantics, hence the metaphysical principle of determinacy and the ensuing optimism places a heavy burden on the powers of the human mind. To be blunt, it is fantastic that the mind should be that powerful. Consider the following form. This is actually a short style. So consider the following form, which is this sort of AE-type form. Yeah, for every x there is a y, there's a unique y such as phi xy. For every x, there's a unique y, y, x, and y. I call this ae, actually ae to phi. I assume for the sake of definiteness that the display variables range over natural numbers. We saw above that Dodd's arguments concerning language acquisition and understanding point toward Haydn semantics. So under Hating's Semantics, a canonical proof of a sentence of this form, AE, consists of, again just to spell it out, but it's here, yeah, a procedure that, given any natural number N, produces a construction of a, oops, given any natural number M, that should be, produces a construction of a natural number N and a proof of phi MN. So under hating semantics, the form A-E amounts to the existence of a computable function f, such that for every natural number n, phi of n and fn. That is, for every x that is unique y phi xy is a statement that a certain function is computable. This, at any rate, is how the Domenian anti-realist understands the form A-E phi, right? That there's a certain function that is computable.

42:30 You just need to follow through the Haiti definitions and that's what falls out. Church's thesis is the statement that every computable function is recursive. It is generally agreed that in the context of classical mathematics, Church's thesis is not equivalent to any formula or scheme in the standard axiomizations. Some deny that Church's thesis is a mathematical matter at all. The reason for this, of course, is that computability is an informal notion. However, we just saw that in the context of Haydn semantics, computability has a definitive formulation via the form AB. So, in Haydn semantics, at least, we have a rather definitive formulation of what we mean by computability. So, for the intuitionists, Church's thesis is a scheme in the language of arithmetic, which is at the bottom of this slide. So, if for every x there is unique y phi xy, then the thing on the right is just writing out that there's a recursive function that computes the y given the x. Again, so we don't need to go into the details here, but that's, in effect, what it says. So if there's a computable function, that's what the left says. On the right, it says there's a recursive function. It does the same thing. All right. Among those inclined toward taking semantics, the issue of Church's thesis amounts to whether CT is an acceptable principle and whether it should be adopted as a new axiom scheme. Now, the scheme CT is consistent such as hating arithmetic, again, this was a very powerful discovery, you know, 30 years ago or so, that you can add CT to hating arithmetic and it's still consistent. So then there's a debate among these tuitionists as to whether it should be accepted or not. And that's really how the debate of Church's thesis is focused in that camp. But CT is inconsistent with classical arithmetic, and in particular with excluded middle. So, to see this, let phi xy be the following formula. I think this is on the next slide. It's the penultimate slide. So, we're up to the penultimate slide of the penultimate speaker. So, let phi xy be a... Oh, all right, be the following formula. thing on the left says is that the Turing machine with girdle number x when given x's input halts and y is 0 or it doesn't halt and y is 1. So phi x y says that either the Turing

45:00 machine with code x halts when given x's input and y is 0 or else the Turing machine with code x does not halts when given x's input and y is 1. Now consider the following instance of excluded middle, right? Exist X, Z, T, X, X, Z, or not? Yeah, that's up here too, right? So this little proof is on the slide and it's also on the handout. But it follows, all right, it easily follows that for every X there is E equal to Y such as psi X, Y. All right, follows from excluded middle of this A, E of this psi. All right, so, but it follows from the unsolvability that is that you could prove even in an intuitionistic arithmetic that the halting problem isn't recursive, that there is no recursive function that computes the y given the x. That is we can show even in an intuitionistic arithmetic that there is no recursive function, and again I just wrote it formally down there, it computes the y given the x. So with excluded middle, we deduce the antecedent of an instance to cd, we deduce an instance of the end of C and refute the corresponding concept line. So this instance of CT is outright refutable in classical arithmetic. Or in other words, this instance of CT is inconsistent with the indicated instance of excluded middle. So no classical mathematician can accept the scheme CT. No classical mathematician can. Of course, for someone who refuses to accept hating semantics, so for a realist now, the philosophical realist, the scheme CT does not express Church's thesis, because for every X there is a unique Y doesn't say that the Y is computable. However, Tennant's classical mathematician does accept hating semantics as an anti-realist. So her rejection of CT does amount to a rejection of the intuitive, informal version of Church's thesis. As we saw in the Hating interpretation of the logical terminology, the intuitive reading of Church's thesis leads straight to CT, leads straight to this scheme here. To reiterate, then, the classical mathematician proves that for every x there is e, e, y, psi, x, y. The Demetian arguments indicate that the Hating understanding of the connectives and quantifiers recapitulates the only meaning they have. So according to that meaning, that we can prove that we can compute the y given the x.

47:30 Thus, the Dometian who finds a way to accept 10's principle of determinacy as synthetic a priori in order to hang on to classical mathematics must establish that there is a computable function that decides the halting problem. This despite the fact that no recursive function does this and you really have no idea how to compute the function. by itself the metaphysical principle of determinacy does not provide the requisite procedures it doesn't tell us how to compute this function it just assures us that there is a computation the mere adoption of a principle does not justify those consequences that just by adopting a principle you can't get that there's a computable function it seems that Tenet's classical mathematician is saddled with a transcendental non-existence existence argument that is the existence of this procedure that decides the halting problem instructivism. Perhaps the metaphysical principle or the ensuing optimism provides some assurance that the computation-free procedures exist. Even so, this will not do. On the Haiti interpretation of the existential quantifier, our theorists cannot assert that the requisite procedures exist unless she knows how to find it. The theme of antirealism is to rule out such unknowable procedures. The situation is quite general. I think I'll skip over this, but it doesn't matter that we're focusing in on RE, right? Any arithmetic predicate, right? It ends up having to be decidable by essentially the same argument. All right, last section called We Cannot Run. Let us sum up, again, as you said, we're almost done. Define modesty to be the thesis that there may be some predicates of natural numbers that we cannot effectively decide, right? So modesty, again, is a See, the last slide adds, oh yeah, that's the last thing's on it, as it should. Okay, that's it. So, to find modesty to be the thesis that there may be some predicates of natural numbers that we cannot effectively decide. Not saying there is, just saying there might be. Modesty is just the statement that, for all we know, there very well might be at least one well-defined predicate chi of X, such that there is no effective procedure that decides of every natural number N whether or not chi of N holds. Modesty is just the statement that we have not established the truth of optimism. That's all. And so it is modest indeed. Church's thesis is the sharp version of modesty,

50:00 but since it entails that there is no non-repressive predicate, it is effectively excitable. And we know that there's plenty of those. predicates. Tenet's principle of determinacy underwrites excluded middle and as we have seen, excluded middle is inconsistent with modesty in the context of making semantics. One who claims to know excluded middle, a priori or otherwise, thereby claims to know the truth of optimism and thus the falsity of modesty. Despite the fact that we have no clue as to how to effectively decide some predicates, like the halting problem. One who even assumes LEM, thereby assumes the truth of optimism the falsity of modesty. The only way for a Dometian anti-realist to endorse excluded middle is for the discourse to be decidable. In a rich context like that, or arithmetic, this requires gross immodesty, or what Helen calls hubris. To conclude, then, let us briefly introduce Domet and Tennant's philosophical opponent, the realist, the philosophical realist. I'm separating the classical magnification, of course, from the philosophical realist. This philosopher can hold that there may be unknowable truths. He rejects the anti-realism. There may be unknowable truths. And that there are or might be predicates that humans cannot effectively decide. That modesty is consistent with realism. However, since he rejects hating semantics, there is nothing immodest about his adoption of excluded middle. Rightly or wrongly, coherently or incoherently, our realist presupposes a semantics that the truth can outrun noability. In claiming that excluded middle is true or known, our realist is not thereby claiming that every sentence is either knowable or refutable, nor that every predicate is decidable. Again, for the realist, philosophical realist, truth is one thing, knowability is something else. It is the main conclusion of Thummett's arguments for manifestation, separability, and harmony that this distinction is untenable. that is, the distinction between just is, no ability, a principle. So if the classical mathematician maintains Church's thesis or some other semblance of modesty, he cannot abide down its arguments concerning semantics. In particular, he must reject the anti-realist meanings assigned to the logical terminology. That is, if our classical mathematician maintains modesty, he cannot accept Hayden's semantics and cannot go along with Tenet's grand synthesis as friendly as that may look.

52:30 The problem here is not with the metaphysical principle of determinacy, which Tenet saddles the realist with. The principle that the world is determining, the mathematical world is determining. Presumably, our realist accepts that. The problem is with the combination of determinacy, or excluded middle, and hating semantics. Our conclusion cuts both ways. From hating semantics, the natural outcome of Domet's arguments for anti-realism, we see that excluded middle and optimism and immodesty are intimately bound together. So if the dominion anti-realist maintains Church's thesis or its formal statement CT or some other semblance of modesty, she cannot have any truck with excluded middle or any metaphysical thesis that underwrites excluded middle. The anti-realist who endorses modesty has a principal reason to reject the truth of some consequences of excluded middle outright, not just to demure from its assertion or its status as a logical truth. the conclusion cuts both ways. The Dometian, who maintains modesty, or churchisties, it's a sharp form of modesty, now has a principled reason to outright reject excluded bill, or at least in this form, rather than just to demeure from it. Against Tenet, we thus cannot have our cake and eat it too, no matter how polite we are prepared to be when doing the mathematics, or when eating it, as he says. If one is determined to accept classical mathematics and still maintain a sense of modesty, then he must reject thesis that Hattie Semanics provides the meaning of the logical operators of this classical mathematics. Consequently, he, the classical mathematician, must reject that its arguments leading to anti-realism either out of hand or by finding some fault in the reasoning. Thank you. I wonder if I can the dialectic theory debates, and that seems to be that there's a potential rapprochement between magicianistic mathematicians and political mathematicians via this intermediate view, which is the optimist. I don't know if it's mathematicians, it's there which actually is non-interviews, anti-realists. So the anti-realists will retain heightened semantics, as of the optimist, and the optimist logic in addition to... Yes, according to Tenet

55:00 well, we don't have to do this that's the only, listen by an infestation and so that's it's sort of a rapprochement because there's some shared principles there between the two of those encounters shared meaning but Tenet puts it in the form of this metaphysical principle of determinacy that the classical application adopts The argument on the paper is that the metaphysical principle of determinacy entails optimism in light of intubulacy. Right, okay. My thought was two thoughts. One is that optimism might just be a nonstarter because of paradoxes of nobility. and the second thought was I was worried about your you wanted to reduce I think I read you right you wanted to reduce optimism to absurdity by saying there might be absolutely undecidable oh no, not reduce, I just want to show where optimism goes right but if there were absolutely undecidable propositions then optimism would be false then optimism would be false that was part of what came out of your play yeah, that's what Pearl pointed out had a logic than any logic in 1931 where he wanted to reconstrue hiding semantics to allow for absolutely undecidable propositions. That was a way of admitting for a weaker intuition which allowed true value gaps, so you could have generalised optimism. But what about classical language? Not really, but you'd still have a form of optimism which still admits for absolutely undecidable propositions such as the ones that you alluded to. Well, optimism that's undefined Right, I can allow for that by saying a weaker form, by saying, so saying it's either provable that P, or it's provable that not P, so it's either provable that P is true, or provable that P is false, and so you have, in the same way which a truth value gap person might deny by valence, and so you can weaken optimism, so it's compatible with these absolute undesirable propositions, how are this weaker logic? That doesn't seem to fit the Dometean framework, where a falsity just is, right, without provability. Well, there's no reason why any Jewishness shouldn't go that way and weaken his logic to an amateur.

57:30 If I think Gap's to accommodate all sorts of phenomena, such as absolutely insightful propositions. Of course, I'm just thinking of mathematics, sir. Right. Well, Heighton thought there might be absolutely insightful propositions in the mathematical world, too. That's why he was pressed this way to do well. Otherwise, we'd end up denying LEM directly. Well, the denial of an instance that would exclude middle forces is contradictory. Yeah, and that's what Heighton wanted to do. He wanted to try to be able to deny it in some sense. Exactly, to make some that so that it could turn out that particular counter instances. So, just because it'll be the first one about just wording out optimism by a pitch, just getting rid of pitch and saying... the intuitionist has that, sorry, the anti-realist has that problem too. I mean, if all truths are knowable, then somehow the intuition, the yeah, let's get a little strange terminology. There's two philosophical views and two mathematics on the table. I'm wondering how they fit together. So on the philosophy side, there's realism and anti-realism. And the thesis of anti-realism is that all truths are knowable. And then on the mathematics side we have a classical and classical and intuitionistic. the anti-realist is going to potentially have switched by problems anyway. It has to deal with those. The anti-realist says that all truths are knowable. What negation means, the anti-realist in that sense can cope with pitch. Perhaps, yeah, but I know the tenet at least takes it quite seriously, so he must see this, he's got it, he's got it with all this. So Fitch-type issues have to be resolved, you know, at least according to tenet. Anyway, whether or not you accept the exclusionary model. But you're suggesting, well, maybe there might be some way to resolve them which is inconsistent with the optimist, something like that. Yeah, just knock out any sort of rapprochement between realists and anti-realists by Fitch. Assuming that the anti-realist has a solution to the Fitch problem. Exactly. It's inconsistent, yes. Yeah, I suppose that's possible. I haven't explored it. Is there a question? I'm afraid I said I'm taking any more of a disagreement.

1:00:00 I'm easy about something. I'm not sure how it impacts on the rise of the structure that you have had. The little idea is in green. It takes us from the middle to the claim of the US of the side of the issue. After we get to the conclusion, what, if we're intuition, so I'm not really smart, are we supposed to think? How do you unpack the thought that we shouldn't accept the student level? If it goes like this, if the thought is It's a kind of contraposition, right? So you've had the thought, so if I accept the pseudo-middle, I'm committed to asserting that an arbitrary proposition is null, or it's negation is null. Right. Okay. But that needn't be so, plenty of cases where I can't guarantee that. Okay, so I shouldn't... I should do the middle. Yeah. But the thought that needn't be so is already inconsistent with epistemic constraint. Right. That's the thought that, well, it could be for a particular selective phi, that fires a noble and neither does it's negation of those. Contraposing back gets your contradiction. No, we're not obviously getting, we always say can't transpose back because the negation of this is going to be a... Quite, that's what I'm saying. Okay, so what's the appropriate way of expressing the caution that invites the contraposition? Well, we have to de-ear from, well, so final conclusion would be this one. Presumably we want to de-ear from that unless we're going to be a lot of optimists. What does demurring from it mean? What's the thought I have when I demur from it? Don't accept it. No, it's something in truth. It could be that it's not true. Well, no, that can't be. No, that assumes excluding. No, we'll put it that way. For this sense. That is, you're assuming that this sense is true or false. To do your firm, it can't be to assume that it's this. No, no. To have a thought, it could be that something isn't true. That is, to accept excluding. No, no. Acceptance. Look, it's true. The thought is... No, no. The viewing is not the same as accepting that it might not be true. He goes, look at this one here, right? They're going to want to do your own excluded middle, but they can't accept that it might not be true. Right, that's my point. So what is this notion of demurable? Well, it's me. I mean, that's what the intuitionist has to come up with. But it can't just be, oh, well, I'm going to refute this. I'm going to deny this, therefore I can deny this. Obviously, that can't be right. But it still is sort of like a reasoning. If I accepted this, I have to accept this.

1:02:30 And I'm not entitled to that claim. Right, I'm not entitled to this, so therefore I'm not entitled to this. I'm not entitled to the claim even though I can exclude the negation. I'm not entitled to the claim even if what? Even though I can exclude the negation. Yes, that's right. Yeah, because, again, not I. But this is supposed to be an argument for rejecting student middlemen. Not rejecting. Yeah, rejecting. Oh, oh, oh, right. Yeah, I actually claim that. you're getting an informal idea of why someone who believes an epistemic constraint might be queasy by the student middle and the argument is well at least the consequences about which one should be queasy but you can't explicate the queasiness about the consequence without presupposing your data set level negation with an F because you do know the negation of the concept of fails given an epistemic constraint so it looks like the argument already working the meta-logic in which D&E fails. Which is classical. Oh, where D&E fails, of course. It's supposed to be an argument for Scuderbiddle to fail, and that's the same question. Not fail. Besides the optimism... This doesn't turn on fail, do, no, reject, deny. The point is different to those distinctions, whatever they are. Let's try again. Sorry. The argument is very intuitive to that. If you accept Scuderbiddle for arbitrary five, you're committed to universal And then the thought is, that's something that's entitled. Not only that, but it's wildly plausible. Go ahead. That's the same thought. I can't guarantee universal decidability. So I shouldn't accept I'm committing to it. I'm focusing on this lack of a guarantee of universal decidability. What is it that I can't do? Universal decidability says, Phi is noble, Phi is defutable. I can defute the negation of that disjunction using principles I accept so one thing I can do is rule out its negation but I don't say nevertheless I'm not entitled to it I must already believe in a distinction from double negation and the claim which it is double negation if I believe in that distinction I don't need a citation go ahead and search it shared content again I must already be committed to deny to demurring from refusing to accept whatever double negation in the meta language if that's my previous in order to run the argument. So I don't need an independent

1:05:00 argument that will obviously the middle fail. That's the same thing. Yeah, so it's a problem of sharing contact again. In the middle language we have to be using intuition. No, but we've already Yeah, we've already just kind of put this, we've already decided that. I am an intuition. I'm not bothered by sharing contact with anyone. Yeah. Okay. I'm trying to recover the train of thought that makes me as an intuitionist unpersuaded of the student middle. Yeah, you're nervous about it. And I find I need to presuppose I already reject obligation. Yeah, because as an intuitionist, you do reject it. It's not logically true. It's not necessary that an argument in the meta-language is the same strength as an argument in the formal language. You could reject or accept a principle in one language but not in another, because they may be in your mind referring to different concepts. That's perfectly true. The question is how to reconstruct a rationale that for the first time moves someone who believes in epistemic constraint to a doubt about his community. We've already agreed that... I'm saying that the argument that we get here, it looks like it does it superficially, actually begs the question. It only goes through if there's already a reservation about obligation and limitation. If there's no reservation about that... according to we all have to agree already that double negation of elimination is not logically true I think what's this argument for? I think institutions do not deny the law of excluded mental, absolutely they do not accept it everywhere and it may be possible but in different languages it may be acceptable in different contexts but in Tenet's framework we've already agreed that supposedly that double negation of elimination is not valid logically and excluded middle is not valid logically he said that excluded middle is this metaphysical principle this is Tennant's dialectic excluded middle is not true logically I don't care about Tennant's framework I want to know what is the intuitive thought that you presented to us that's supposed to explain why these guys don't accept the fluid middle no it was meant to be a reductio in part reductio in ten Actually, we may do a bit better than this. With the church's thesis example, we actually get an outright reputation.

1:07:30 Oh, maybe I should do this thing on the right. I think we're finished with this. Because people like that better, apparently. Besides, I'm tired. I'm pretty bothered by this light. So we have that, let's see, so CT, now, if you don't like CT, you know, if you don't like church's thesis, you could put in something stronger than that, like, you know, but we have CT, and then we assume excluded middle, and we get an outright contradiction from this. Now, the instance that would exclude the middle is not of this form. Well, it is like this, but it's not where phi is a sentence. There's a free variable in there. So what we end up with is that CT entails, it's not the case, outright entails, that it's not the case that for every X, phi X for some particular phi or not phi X. So it's not just a general, an outright rejection. So if you accept CT, which is a form of modesty, or some other even weaker one, It could be that, if you don't like church's thesis, it could be, you know, something a bit quicker than that, or a lot quicker than that. We actually can reject this, and then we can do a, yeah, and then we can do a, so if you accept that this, you have to outright deny CT. So we don't need this play with the neural and the like, at least for example like this. I don't know if that helps, but at least it takes us. You know, I need some unpacking, though. What about the sound of the original simpler argument? This one doesn't involve having to denure or... I mean, the original argument I gave was in the context, I was assuming for the sake of argument, that Neil was right and that we are working in a background intuition-stick, you know, metaphorical. I wasn't trying to make any brain cleanse about how we're supposed to... No, but my objection was, with the green argument, that you need to have the distinction between double negation to the regular topic, which is basically... Yeah. Otherwise, the argument is said about this. Yeah, the argument destabilizes from a, if you're, yeah. Now, here we need to have a distinction in place to not for all x and the x not, then. What do you mean the distinction? This time we get an outright interpretation of it. Yeah, we're going to accept not for all x, right? No, we can have, we can have, it actually, if you accept CT, you have to actually deny, deny this, not just. Deny the negation? Deny this, right. For every x, phi x or not phi x, for that particular phi.

1:10:00 Not just demure from it, but deny it. Okay, and the thought is, what do we do with that? Well, we were in trouble with the mural. I don't know if you weren't, you said it didn't matter. I understand the problem with shared content, and I don't have a solution to it. I don't think there's a shared content issue here. It is? I don't think so. I'm not trying to persuade a classical mathematician to it, but I'm trying to understand the train of thought needs some who have already synthesized them with intuitive semantics in some kind of encoded way. I just think he should worry about the law of excluded middle. It looks to me like he needs to be already persuaded that double negation is distinct from defamation. Yes, yes, he is. By accepting hating semantics, he already sees that, that those aren't logically equivalent. Well, then he doesn't need this argument. He doesn't need this argument. It's the same principle. Right. That is, he doesn't need the argument to be convinced that law of excluded middle is not logically true. But he may hold that it's true on, you know, this is within the Tenet's framework, right? he might want to hold that it's true anyway. It's just not logically true. And if he does hold that, then he's not a very, I think he has a wildly plausible, you know, he's attributing wildly implausible epistemic powers to the human from that, from the hating framework. So I wasn't, you know, this isn't supposed to be, from the, I was assuming at least for the sake of argument there, that we've accepted that hating semantics is the correct one. and the excluded middle isn't logically true. So Tenet says, that's what Tenet says, it's not logically true, but the classical mathematician can accept it as a metaphysical principle, as a synthetic one. And then my diagram, then I would point out that however you accept it, whether it's logical or not, it follows from that, that every sentence is either knowable or refutable. So the person who accepts the metaphysical principle there's a sort of wild optimism. But it doesn't seem like you should go both ways. I mean, it seems like it's a necessary condition of optimism that you should believe in determinacy, but it doesn't seem that determinacy, which is in a physical principle, would entail something epistemic like optimism. It does in the context of... It does in the context of having semantics. So for anything real, it's to advise this. It goes both ways. It goes both ways. That's why he's trying to argue again for the classical mathematician at the end, right?

1:12:30 Sorry, the realist at the end, right? He said, no, truth is one thing, no, but at least something else, right? He'll accept the determinacy, and he's not going to rely on hidden semantics. Okay, one more step. Why is the optimism wild? Well, because we have no reason to believe that. But we do have reason to include this negation. Yes, that's right. So in order to maintain both those points of your simultaneity, I must already believe in distinction between allegation and affirmation. Yeah, I'm not denying that. Of course you do. Of course there's a distinction. But if I accept that distinction already, I don't leave the argument. It's not unpacking the motivation. There must be some other motivation for having that logic. The motivation for having intuition as a logic is presumably duds, right? You read that and you buy it and you say, OK, this is the logic we have to use. But this is meant to be an argument for not accepting the excluded middle, isn't it? It's meant to be an argument for not accepting the excluded middle logical principle as a synthetic principle from that point of view so there are two arguments is that the point there's the argument you haven't told us about yet which which generates a doubt about excluded middle and doubling action in the match in the logic yeah it's being assumed that excluded middle is not logically true it's non-analytic like the tenant says that the classical application you know we can still be friends he shouldn't think it's logically true but he accepts that it's true synthetically I guess you need to follow that right but he accepts that there's a logical distinction between not not I thought you looked like you were ready to help your I thought I understood what you were saying Tenant is it's embarrassing to just repeat it Tenant thinks that something like the manifestation argument adopt semantics, and so that doesn't validate the law of excluded middle. So you've not got that anyway. The reconciliation with classical mathematics is to say, well, you can't accept the law of excluded middle as a logical principle, but you could believe it as some kind of metaphysical principle. But my reservation is not sophisticated. Just a bit. And then Stuart is arguing, well, not even that's any good, because even if, never mind the status of phi or not phi or metaphysical principle. Whatever sort of status you've had, it leads to an optimist, like a solid middle. Okay, fellas, here's the question.

1:15:00 Can you unpack the gap this place to open up, on roughly speaking, heightened semantics about a student middle, without as an effect running something like that green thought? Why doesn't heightened semantics validate a student middle? Good question. Yeah, I'll take that. Now, I think what you'll find when you think about that is you run, in effect, the green thought, the green eye. Yes, yes. Then my point is the argument circular. Okay. That's right. So I was just taking it for granted that excluded middle... So we haven't established, which I was just taking for granted, so you're claiming we haven't established that even on hitting semantics, that non-naphi is invalid. Right. Or that excluded middle is logical. We haven't shown that it's not. not only that but in order to articulate this thought and say why there's a gap you need to presuppose a distinction yes and I was just assuming that that's already done because that's what my opponents I can concede that to my opponent in this case to the anti-realist or the reforming anti-realist so if for some reason he can't establish he can't establish that if he can't establish that this is not a law without assuming an equivalent resolution then there's nothing to reconcile so I was just taking for granted that okay I'll grant that that this isn't lawfully true I'll grant it not I started off by saying I have a worry I'm not sure it impacts on your argument or not there seems to be something unstable in the state setting he says that there's a difference, right, between not on P and P, and I say, okay, I'll grant that for now. And then I show that it leads to this implausible conclusion. And then you say, oh, but look, why shouldn't even grant that for the sake of argument? Am I understanding what you're... No, it's only shouldn't have granted the sake of argument. It's rather that you want to understand what's implausible about the conclusion. From Tannin's point of view, you need the distinction between double legation and affirmation. But that's the very distinction which the argument is meant to motivate. Not this argument. I would just take that distinction for granted. My argument isn't meant to motivate that distinction.

1:17:30 I was just assuming that if there's no distinction, then there's nothing for me to argue against. So are you suggesting that maybe there is room for the anti-realist? so if the anti-realist I'm saying that we haven't so far understood the basis of the anti-realist modesty it seems to be some tension in the standard way of explaining it until we've understood that it's not clear what it would be to metaphysically assume something consistent with the modesty it's our clarity about the whole dialectic can't you be modest without making the question about the location you just say there's no present warrant to assert to the optimist There's some evidence for optimism in the case of the Four Cold Problem, but there's not conclusive evidence, so it's not presently known whether everything, either a four-propositions around the proof will, or the allegations are proof. And that's a stance which doesn't motivate the honesty. it is known but it's not the case so we already need actually I don't know if that's right it's not the case that there is so it's not So we don't know that it's all x, phi x all, not phi x, okay. Oh, yes. Right. This isn't known. Where phi x is now, it's x a proposition and phi from phi is decidable. Sorry, phi is provable. So it's not known that it's all x, either phi x is provable or all. Yeah. It's not. Yeah, it's in schematic form, right, or something like that. but we do know not e x naught ok, on this intuitive view we do know there's no count examples of that instruction so we're already postulating a contrast between 4x and not e x naught with respect to the same matrix that's a distinction that's only intrusively intuitionistically it's not a personal distinction so we must already be intuitionist and running this argument I still think that in the dialogue

1:20:00 So if he can't establish that the fire I find is not logically true, then he doesn't have a reason for reforming in the first place. That's right. That's the point. So what's the motivation? Well, that's, you know, retentive, right? Detaining of the true or whatever, and see if there's anything in there, right? But I'm just assuming that he has reason to reform. And then he says, look, we have a reason to reform the logic, but here's how to get classical mathematics anyway. and then I say, no, but look how bad that is. And you're saying, well, he doesn't have reason to reform in the first place. Or might not have, or he hasn't shown that he has. The question is, can he have a reason to reform in the first place? And also, take your point, the situation is bad. If you've got to stop the classic logic in the way you're describing. Yeah, not that simple. It's a little step where you understand both points, right? The original reason to reform. We're not helping ourselves to classical logic, we're helping ourselves to classical mathematics. Because we're not going to make this logically true all of a sudden. We're just going to accept it as a non-logical. Okay, but you do need, for your argument to write, you need the following combination. You need to get the case that there is a good prime facial motive for logical reform. Okay. And then, of course, the queasiness, when one sees the implications of going classical. After you've made the reform. After you've made the reform. That's right. That's what I have a doubt about. You have a doubt about? About the stability of that combination. the combination of so once you go intuitionistic in the logic then there doesn't seem to be do you lose any grounds for performing as it were it's very difficult to see what if you if you're accepting intuitionistic logic and then you add in Well, maybe not. I mean, it's not clear what the original problem was. I'm not saying the intuition should accept classical logic, right? No, no, right. The question is what the official account is of the grounds for not doing so. For not accepting intuitive logic. Not accepting classical logic. Not accepting classical logic, yeah. The grounds would be... And it looks like the best state we have so far is circular. best statement of their view that we should reform it. Of the grounds for it, yes. For reforming, right.

1:22:30 Right. Now, you then wind up with an argument that if they were known to have an add-on, except excluded middle and other classical principles, they get huge epistemic overcommitment. We never understood the original sense of epistemic overcommitment. That's what wasn't explained. Even in the simple argument. and the problem with it was that it seemed to presuppose a distinction which was supposed to motivate effectively the distinction which I was conceding for the sake of argument that that distinction is okay so if we don't understand what the intuitions of the reservation want there, what the sense of overcommitment was in that original simple case how can we be confident that there's an overcommitment here in the cases that you can develop but it follows Well, why is it an overcommittal? That it follows that we have an overcommittal? Classically, it's an overcommittal. If you understand what this means classically, we shouldn't be making this play. We can't justify it. Classically, it's not an overcommittal at all. Well, I don't mean by classically, but as far as realism goes, this inference is no, but from phi to phi is no. So classically excluded middle is, or sorry, yeah, as far as classical mathematics goes, as far as realism goes, were not over-committed by accepting excluded knowledge. Because they denies, the realist denies that these connectives have an epistemic component. So what do we make of the sharper argument then from Church's thesis then? So we have right, if Frederick... Before we get on to this, has anybody got any more questions? Anything else? I was wondering about the proposal which of course the intuitionist would interpret as saying there's a computable bunch with the property. If you're the sort of optimist that we're trying to refute, would you be forced into interpreting that as meaning that there's Yes, that's the claim. If you accept Hayden semantics, then anything of this form, for every x there is a y, phi xy, amounts to a claim that you have given any x you can find the y.

1:25:00 which is the claim that Why do you say that you I mean, surely you're a motor I mean, there is a construction given in the end not that you have one but that would say every problem is But that existential claim has got to be understood constructively too when you say there is a construction you know how to find one What's the distinction in this reading of that that sentence and the one that you get by inverting quantified. Inverting which quantified? You mean exist x? Yeah. You see if you read it as though the interesting understanding had to be an effect equivalent to the inverse. No, no, there is an f. Does there exist a function? No, there exists a function. Right, so this would be actually the classical math partition generally will accept this one too. There is an f now, this isn't such that for every x 5x and fx. So this is a principle of sex or a logic that everybody makes sense. If for every x there is a unique y, it's unique y. If there wasn't unique, there would be choice involved. That's the choice involved. But if for every x there is a unique y such as phi xy, then there's a function such as given any x, phi of x and fx. So it reverses the quantifier sort of, that this becomes a function when it becomes up front, not the y. Now, if this is understood in an aching way, then this function here is going to be a computable one. My point is simply this, that you seem to be interpreting the upper centers as though it had to be read as the lower, let's say, some uniform operation, which you specify in advance, For arbitrary x, we can use that to find a y-central phi-x-y. Isn't that what the I don't think so, no. I think the hat-in semantics is consistent with a more modest reading. We've recognized in advance that for arbitrary x, we can find a y-central phi-x-y. Remember, this will be a procedure too. So then for every x means that you have a procedure to give in any x gives you the proof of this. Right, but why is that required? I'll read that off the original infomalics, but no, it's for me. This? All right.

1:27:30 The statement for the universal quantifier is, again, this is just almost right out of Haken. This is where we find out. Anybody know which slide it was? It's on the anti-aventure. I see. Move on. Well, I mean, it's 22. Oh, so you want to play it? Okay. Oh, it'll be quick. Oh, all right. So the hating semantics for the universal quantifier for every x-phi is there's a procedure which, given any x, will produce a proof of the instance. So there's a procedure that, given any x, will produce a proof of this. Now, proof of this consists of a y and a proof of the phi x-y. again looking on the and that implication there says not only that there is a procedure it says that the procedure is definable in whatever language and theory you've had already if the metal language is also intuitive that's right but it doesn't matter for now it's a little bit stronger there that was saying that there is just a procedure yeah if the metal language is intuitive it presumably will be All right, so I claim, I thought this was rather uncontroversial, that when you understand this in my opinion semantics, this just is a statement that there's a computable function. And that, and if you, so if you accept Church's thesis, then you'll believe this, and this is outright inconsistent with excluded, though. well some do and some don't but any it doesn't have to be recursive you can put anything over here it doesn't have to be recursive but any will be out this is actually a point that David McCarty made this deed that, and I think he might have been tracking something very similar to your worry, that we still, there's still a gap in the intuitionist argument, because, against the excluded middle, we still don't know what its status is, right? The other, the other intuitions, the hindrance has just all seemed correct, when we come to excluded middle, again, we don't know whether that's correct or not. So we're looking for an actual ground to, for the, for the hating,

1:30:00 anti-realists to actually reject excluded middle and if Church's thesis is one such crime so again it doesn't seem like any questions are begged here with this one if you accept Church's thesis in this form then you can't then excluded middle is you know the stronger version of excluded middle is inconsistent like this one Thank you.