TheSim

The conclusions drawn by the scientific method are independent of one's definition of personal identity. Therefore science has nothing to say about the truth of mortality or immortality. Instead we must instead turn to conceptual analysis, where it becomes clear , at least to my mind, that whatever one's conclusions are regarding the truth of mortality or immortality, one's conclusions are nothing more than a restatement of one's definition of personal identity.

Sure there is no bijection between a set and it's powerset in the case of finite sets since this result can be directly constructed. But Cantor's theorem is a so-called 'proof' by way of contradiction that begs the law of excluded middle without right in order to lead to his desired conclusion. He seems to argue that since N -> P(N) cannot be a surjective function without leading to a contradiction, that it must therefore be the case that N->P(N) isn't a surjection, thereby concluding that |P(N)| > |N|. But he has no right to do this because he hasn't shown that P(N) is a set. Obviously Cantor's theorem is expressible as a theorem of ZF since the rules of classical logic are permitted there. But as Skolem's Paradox demonstrates this leads to a contradiction regarding its intended interpretation. This is why it is probably better to abandon ZF for a constructive set theory.

Second order logic has certainly been used to create a sentence that has traditionally been called "completeness of the reals", but to 'prove' completeness of the reals in second-order logic requires assuming the very existence of uncountable and non-computable sets of real-numbers that are in question. At best all you have is an empty axiom of syntax called "completeness of the reals" that cannot be meaningfully interpreted. And if non-constructable sets are dropped from second-order logic, it collapses into first order logic. Cantor's so-called "proof" of the existence of uncountable sets is merely a conditional statement saying that if the power-set of natural numbers exists then it cannot be enumerated. But we have no way of constructing this power-set, since only a countable number of subsets of N are recursively enumerable, that is to say, can be generated by an algorithm. Sure, we can define the phase "Uncountable sets exist" to refer to Cantor's proof, but this result is merely an empty and circular parlour game of syntactical gibberish without physical or practical significance. It might be a fun parlour game, but i don't see how it leads to philosophical clarity.

But it HAS been proven. See Skolem's paradox. Every countable axiomatisation of set theory in first-order logic, if it is consistent, has a model that is countable. This result should only be surprising to those who literally believe in non-computable reals.

Perhaps it would solve the philosophical confusion if a line is defined to be a set of ordered constructable points, where a point is a computable total function by at least one terminating algorithm. That way there is explicit clarity that there are no "holes" in our field of entities - except perhaps for the partial functions corresponding to non-terminating algorithms that cannot be ordered by their outputs - and that there is only a countable number of entities describable by mathematics.

Your problems appear to be with regard to the soul being conceived as a substance. But not of all our indispensable concepts are backed by substances.

Universal statements of the form "Every X has property Y" where X has unlimited scope, are best understood to be policies for generating empirically verifiable finite propositions of the form " precisely n X's have property Y", where in order to generate a proposition the variable n must be substituted for any finite number. It is analogous to infinite FOR loops in programming languages. Whenever a programmer uses an infinite FOR loop, he isn't implying that the algorithm will never ever be stopped, he is merely deferring the termination of the FOR loop to the external operating environment of the program and has nothing else to state about the matter. Likewise, so-called universal "laws" of science are better understood to be policies for generating verifiable and finite propositions that we deem to be permissible in light of our current observations. Here, the "stopping conditions" of a policy of science are the fulfilment of falsification criteria for one of it's generated propositions . As with a non-terminated infinite FOR loop, a non-terminated policy of science implies nothing empirical whatsoever, positive or negative, concerning the eventuality of it later stopping due to falsification of one of it's verifiable propositions. Put another way, science is empirically constructive and expresses a finite amount of information concerning what has happened and what can be envisaged to happen, but science implies nothing empirical about what cannot ever happen, despite occasional appearances to the contrary. For example, take the so-called law that "Nothing can accelerate faster than the speed of light". On the surface, it looks as if if it were a meaningful negative empirical statement that universally forbids a genuine empirical possibility. But what it is really saying is that "an object accelerating faster-than the speed of light" is a nonsensical sentence in the language-game of relativity that isn't even a proposition. It is a statement of grammar, rather than a statement of empirical fact.

Is scientific truth inferred via replication, or is scientific truth equivalent to replication? A problem with the former realist conception of truth is that it understands justified true belief in terms of predictive accuracy. But this is problematic, because our identification of predictive behaviour is relative to our cultural conventions. For example, ordinarily we might consider buying a lottery ticket to constitute a lottery prediction. But what has a mere guess concerning tomorrow's lottery numbers got to do with tomorrow's actual lottery outcome? If the answer is "physically nothing" then our pairing of predictions with their physical outcomes is purely decided post-hoc after the outcome has physically occurred, according to cultural convention. Does the realist really want to say that "he was wrong in his prediction" is merely a figure of speech of post-hoc linguistic convention? On the other hand, if we restrict our notion of a prediction to a known statistical correlate of an outcome then we have eliminated scepticism concerning induction, since there is no longer a conceptual gap separating our notions of induction and prediction for justification to fill. We have also eliminated the cultural relativism of convention, since mere guesses are no longer considered to constitute predictions. Yet the resulting conception of objective truth is now trivial, for our predictions are now definitionally equivalent to inductive bias that speaks nothing of the future.

Cases 1, 2 and 3 demonstrate that the public meaning of a noun is structural. The so-called "realness" of a lemur refers to an non-quantifiable set of empirical associations between an open set of stimuli and an open set of responses to said stimuli, as ambiguously and implicitly laid-out by public conventions that dictate the appropriate use of the word. In contrast, one's private meaning of "Lemur!" when shouted in response to whatever one is experiencing, would constitute a behavioural expression which could either be said to be necessarily correct, or to be said to not be truth apt.

As an ex-rationalist I can sympathise with those who mistake habits of thought for a priori truths. But thought experiments are composed of memories and are ultimately a form of empirical simulation. Einstein was only able to perform successful thought experiments in lieu of observing actual experiments.

If science consisted of a universal method of induction, then it's method could be automated using machines, by translating it into a universally applicable learning algorithm that on average would outperform every other conceivable learning algorithm across every possible data-set. But that isn't the case, as Wolpert's No Free Lunch Theorem demonstrates. In order for a learning algorithm to have superior predictive performance with respect to one group of data is for it to have inferior predictive performance with respect to another group of data. Hence science has no "method" of induction. What methods of induction really consists of is simulating some process (the 'simulated') using another process (the 'simulator') , and measuring the similarity of the simulated process to the simulation by using some arbitrary criterion of similarity. If the simulator is a machine learning algorithm, then this similarity measurement is fed back into the simulator so that it automatically adapts so as to make the simulation more closely resemble the simulated. What is considered to be a simulation of one thing by another thing is in the eye of the beholder and whether the simulation is satisfactory depends on his practical purposes.

Consider the boardgame Cleudo. Cluedo's concept of murder is finite and fully decidable; Cluedo's rules of 'proof' always lead to a single and unambiguous conclusion as to "who killed" Dr black that is "true by convention" and obeys the classical logic law of excluded middle. In other words, to have a proof that a certain murder hypothesis cannot be eliminated is to 'know' that the hypothesis is correct. Before the game ends, all murder hypotheses that aren't yet eliminated are equally likely. So it wouldn't make sense for a player to say that they "believe" any particular hypothesis, rather their only substantial beliefs refer to sets of hypotheses. For example, they might believe the culprit was male if there were more male suspect cards still in play. Hence beliefs in Cluedo reduce to sets of observations. Now compare this game to the historical investigation as to who was Jack the Ripper. Here our concept of murder is neither finite nor decidable. For we do not have a finite and closed list of suspects and no matter how much we learn about the past we can never have a controversy-free constructive proof of who-did it. At most we can say that a state-of-the-art simulation of history that we determined through our current and fallible state of knowledge suggests a particular person. At this point, some people will take this simulation to be the very definition of who did it, whereby their concept of truth regarding jack the ripper becomes truth-by-convention as in the game of Cluedo. Others will abstain from accepting this convention and remain open minded. Also, consider a computer game implementation of Cluedo that is played against AI opponents. The computer could either generate the truth of the murder before the game is played, analogous to a realist's conception of history that is factually precise but unknowable. Or the computer might merely generate the cards held by each player, as and when each player requires them, analogous to the anti-realists conception of history as factually imprecise but knowable and determined through current activity.

From a behavioural perspective, all that a person means when saying " There is Donald Trump" are the potential stimuli that can provoke him to say it. If a person's stimulus-responses were completely understood they would never be interpreted as saying anything controversial, even if they asserted that an impersonator was "Donald Trump". Consider how an engineer might react if their version of Amazon Alexa insisted that "Donald Trump isn't the US President". They wouldn't think it was useful to accuse Alexa for being "factually wrong" for they would believe that they had technical insight into the causes of her wayward linguistic behaviour. Rather, they would think there is a database hitch, or that the programmers were playing a joke, etc.

I think you've just given a reasonable description of "absolute truth", namely in terms of facts that are considered to be unchangeable (at least if we are restricting ourselves to physical truth and ignoring metaphysical possibility). But there is a problem here, namely it isn't clear what the object of a belief is or how it should be decided. For example, it is physically true that a Flat Earther has asserted the sentence "The Earth is flat". Furthermore there are physical causes of their assertion. Therefore if the object of the Flat Earther's belief is taken to refer to the causes of his belief then his assertion must be necessarily correct, regardless of the opinions of his community. This implies that the notion of truth is vacuous; any substantial notions called "truth" refer only to matters of linguistic convention. Likewise, if I believe that I know Saturday's lottery numbers but today is only Wednesday, then my prediction has no object unless it is taken to refer to its physical causes. In which case my prediction is again, necessarily correct. After the results of Saturday's lottery are announced I can at least console myself by saying that my earlier prediction is only wrong post-hoc by linguistic convention.

One remarkable tendency of metaphysical realists with respect to a given domain of inquiry is their passion for studying that domain, which they might not have if they didn't believe that there existed an absolute truth to search for. For example, consider a realist and an anti-realist whose attentions are drawn to a dimly illuminated flower whose colour they agree is ambiguous and purple-looking. The realist says "is that really a purple flower? let's turn up the light". The anti-realist replies "Our description is already satisfactory; we agree that the colour is ambiguous and purple-looking. Turning up the light to remove the ambiguity cannot change our current opinion because we would no longer be comparing like for like"