From bef610ff9b87530cde5fe9c997cdea71d85a5d39 Mon Sep 17 00:00:00 2001 From: sajagherri Date: Wed, 10 Nov 2021 00:30:06 -0500 Subject: [PATCH] Minor Edits Small semantic edits, including replacement of "is successful" with "succeeds", and "not successful" with "fails." --- book.tex | 16 ++++++++-------- 1 file changed, 8 insertions(+), 8 deletions(-) diff --git a/book.tex b/book.tex index b21603f..bc34671 100644 --- a/book.tex +++ b/book.tex @@ -662,7 +662,7 @@ \section{Verifiable statements and experimental domains} In the previous section we saw that we can combine statements into new statements. How about verifiable statements? Can we always combine verifiable statements into other verifiable statements? Since all truth functions can be constructed from the three basic Boolean operations, the question becomes: can we construct experimental tests for the negation, conjunction and disjunction of verifiable statements? -The first important result is that the negation of an experimental test, an experimental test that is successful when the first is not successful, does not necessarily exist. Consider our black swan example, an experimental test for the negation would be a procedure that terminates successfully if black swans do not exist. But the given procedure never finishes in that case, so it is not just a matter of switching success with failure. Because of non-termination, not-successful does not necessarily mean failure.\footnote{In this case, the old adage ``absence of evidence is not evidence of absence" applies.} Moreover, it is a result of computability theory that some problems are undecidable: they do not allow the construction of an algorithm that always terminates with a correct yes-or-no answer. So we know that in some cases this is not actually possible. +The first important result is that the negation of an experimental test, an experimental test that succeeds when the first fails, does not necessarily exist. Consider our black swan example, an experimental test for the negation would be a procedure that terminates successfully if black swans do not exist. But the given procedure never finishes in that case, so it is not just a matter of switching success with failure. Because of non-termination, not-successful does not necessarily mean failure.\footnote{In this case, the old adage ``absence of evidence is not evidence of absence" applies.} Moreover, it is a result of computability theory that some problems are undecidable: they do not allow the construction of an algorithm that always terminates with a correct yes-or-no answer. So we know that in some cases this is not actually possible. In the same vein we are able to confirm experimentally that \statement{the mass of this particle is not zero} but not that \statement{the mass of this particle is exactly zero} since we always have uncertainty in our measurements of mass. Even if we could continue shrinking the uncertainty arbitrarily, we would ideally need infinite time to shrink it to zero. What this means is that not all answers to the same question can be equally verified. Is the mass of the photon exactly zero? We can either give a precise ``no" or an imprecise ``it's within this range." Is there extra-terrestrial life? We can either give a precise ``yes" or an imprecise ``we haven't found it so far."\footnote{Note that we are on purpose avoiding induction. It does not play any role in our general mathematical theory of experimental science since the decision of when and how to apply induction violates the principle of scientific objectivity.} @@ -697,7 +697,7 @@ \section{Verifiable statements and experimental domains} \begin{enumerate} \item run test $\expt$ \item if $\expt$ is unsuccessful terminate successfully - \item if $\expt$ is successful terminate unsuccessfully. + \item if $\expt$ succeeds terminate unsuccessfully. \end{enumerate} Since $\expt$ is repeatable and can be executed by anybody, $\expt_\NOT(\expt)$ is also repeatable and can be executed by anybody. Therefore we are justified to assume $\expt_\NOT(\expt) \in \exptSet$. @@ -715,9 +715,9 @@ \section{Verifiable statements and experimental domains} \begin{enumerate} \item initialize $n$ to 1 \item run the test $\expt$ for $n$ seconds - \item if $\expt$ is successful, terminate successfully + \item if $\expt$ succeeds, terminate successfully \item run the test $\expt_\NOT$ for $n$ seconds - \item if $\expt_\NOT$ is successful, terminate unsuccessfully + \item if $\expt_\NOT$ succeeds, terminate unsuccessfully \item increment $n$ and go to step 2 \end{enumerate} The procedure is repeatable and can be executed by anybody therefore $\hat{\expt}(\expt, \expt_\NOT) \in \exptSet$. Both $\expt$ and $\expt_\NOT$ are eventually run an arbitrarily long amount of time therefore $\result(\hat{\expt}(\expt, \expt_\NOT), a) \in \{\SUCCESS, \FAILURE \}$, that is the test will always terminate. We have $\result(\hat{\expt}(\expt, \expt_\NOT), a) = \SUCCESS $ if and only if $\result(\expt, a) = \SUCCESS$ if and only if $a(\stmt) = \TRUE$. We also have $\result(\hat{\expt}(\expt, \expt_\NOT), a) = \FAILURE$ if and only if $\result(\expt_\NOT, a) = \SUCCESS$ if and only if $a(\NOT\stmt) = \TRUE$ if and only if $a(\stmt) = \FALSE$. Therefore $\stmt$ is decidable. This justifies the definition. @@ -734,7 +734,7 @@ \section{Verifiable statements and experimental domains} We introduce decidable statements here because their definition and related properties clarify what happens during negation, but they do not play a major role in our framework. They represent a special case which will we turn to time and time again over the course of this work. -Combining verifiable statements with conjunction (i.e.~the logical AND) is more straightforward. If we are able to verify that \emph{``that animal is a swan"} and that \emph{``that animal is black"}, we can verify that \emph{``that animal is a black swan"} by verifying both. If the tests for both are successful, then the test for the conjunction is successful. That is, if we have two or more verifiable statements, we can always construct an experimental test for the logical AND by running all tests one at a time and check if they are successful. Yet, the number of tests needs to be finite or we would never terminate, so we are limited to the conjunction of a finite number of verifiable statements. +Combining verifiable statements with conjunction (i.e.~the logical AND) is more straightforward. If we are able to verify that \emph{``that animal is a swan"} and that \emph{``that animal is black"}, we can verify that \emph{``that animal is a black swan"} by verifying both. If the tests for both are successful, then the test for the conjunction succeeds. That is, if we have two or more verifiable statements, we can always construct an experimental test for the logical AND by running all tests one at a time and check if they are successful. Yet, the number of tests needs to be finite or we would never terminate, so we are limited to the conjunction of a finite number of verifiable statements. \begin{mathSection} \begin{axiom}[Axiom of finite conjunction verifiability]\label{ax_verifiable_AND} @@ -755,7 +755,7 @@ \section{Verifiable statements and experimental domains} \end{justification} \end{mathSection} -Combining verifiable statements with disjunction (i.e.~the logical OR) is also straightforward. To verify that \emph{``the swan is black or white"} we can first test that \emph{``the swan is black"}. If that is verified that's enough: the swan is black or white. If not, we test that \emph{``the swan is white"}. That is, if we have two or more verifiable statements we can always construct an experimental test for the logical OR by running all tests and stopping at the first one that is successful. Because we stop at the first success, the number of tests can be countably infinite. As long as one test succeeds, which will always be the case when the overall test succeeds, it does not matter how many elements we are not going to verify later. But it cannot be more than countably infinite since the only way we have to find if one experimental test in the set is successful is testing them all one by one. Therefore we are limited to the disjunction of a countable number of verifiable statements. +Combining verifiable statements with disjunction (i.e.~the logical OR) is also straightforward. To verify that \emph{``the swan is black or white"} we can first test that \emph{``the swan is black"}. If that is verified that's enough: the swan is black or white. If not, we test that \emph{``the swan is white"}. That is, if we have two or more verifiable statements we can always construct an experimental test for the logical OR by running all tests and stopping at the first one that succeeds. Because we stop at the first success, the number of tests can be countably infinite. As long as one test succeeds, which will always be the case when the overall test succeeds, it does not matter how many elements we are not going to verify later. But it cannot be more than countably infinite since the only way we have to find if one experimental test in the set succeeds is by testing them all one by one. Therefore we are limited to the disjunction of a countable number of verifiable statements. \begin{mathSection} \begin{axiom}[Axiom of countable disjunction verifiability]\label{ax_verifiable_OR} @@ -920,7 +920,7 @@ \section{Theoretical domains and possibilities} Note that we are closed under countable operations and not arbitrary (e.g. uncountable). Therefore there could be statements that can be constructed from verifiable statements that are not even theoretical statements. One such statement, for example, could be constructed given a set $U$ of possible mass values for a particle that is uncountable, has an uncountable complement, and where the elements are picked arbitrarily and not according to a simple rule.\footnote{Mathematically, we are looking for a set of real numbers that is not a Borel set.} The statement \statement{the mass of the particle expressed in eV is in the set $U$} can only be tested by checking each value individually. But since the set is uncountable and a procedure can only be made of countably many steps, it will be impossible to construct a test for such a statement. -A theoretical statement, then, is one for which we can at least conceive an experimental test. This may not always terminate if the statement is true or it may not always terminate if the statement is false, but at least we have one. The statements that depend on the experimental domain but are not part of the theoretical domain do not even hypothetically allow for a procedure, regardless of the fact that it can terminate, and therefore we do not consider them part of our scientific discourse, even theoretically. +A theoretical statement, then, is one for which we can at least conceive an experimental test. This may not always terminate if the statement is true or it may not always terminate if the statement is false, but at least we have one. The statements that depend on the experimental domain but are not part of the theoretical domain do not even hypothetically allow for a procedure, and therefore we do not consider them part of our scientific discourse, even theoretically. In general, given a theoretical statement $\bar{\stmt}$, we would like to characterize what experimental test can be associated to it. Ideally, we want the experimental test for that statement that terminates, successfully or unsuccessfully, in the most cases. Consider all the verifiable statements that are narrower than $\bar{\stmt}$. If we take their disjunction we get the broadest verifiable statement that is still narrower than $\bar{\stmt}$. We call this the verifiable part of $\bar{\stmt}$, noted $\ver(\bar{\stmt})$. Testing $\ver(\bar{\stmt})$ means running the test that is guaranteed to terminate successfully in the broadest situations in which $\bar{\stmt}$ is true. In fact, if $\bar{\stmt}$ is itself verifiable then $\ver(\bar{\stmt})$ will be exactly $\bar{\stmt}$. @@ -928,7 +928,7 @@ \section{Theoretical domains and possibilities} To each theoretical statement, then, we associate the experimental test constructed by returning successfully if the test for the verifiable part succeeds and returning unsuccessfully if the test for the falsifiable part succeeds. We will not be able to terminate if either of those doesn't terminate, which will correspond to the statement $\NOT \ver(\bar{\stmt}) \AND \NOT \fal(\bar{\stmt})$ being true. We call this the undecidable part of $\bar{\stmt}$, noted $\und(\bar{\stmt})$. -In light of this, consider $\stmt=$\statement{the mass of the photon is rational as expressed in eV}. It is the disjunction of all possibilities with rational numbers, which is countable, and therefore is a theoretical statement. Since we can only experimentally verify finite precision intervals, each verifiable statement will include infinitely many rational (and irrational) numbers. Therefore no verifiable statement is narrower than $\stmt$ and therefore $\ver(\stmt) \equiv \impossibility$. But for the same reason no verifiable statement is incompatible with $\stmt$ and therefore $\fal(\stmt) \equiv \impossibility$. Which means $\und(\bar{\stmt}) \equiv \certainty$. This means that the experimental test for $\stmt$ will never terminate either successfully or unsuccessfully. We call this type of statements undecidable as we will never be able to experimentally test anything about them. +In light of this, consider $\bar{\stmt}=$\statement{the mass of the photon is rational as expressed in eV}. It is the disjunction of all possibilities with rational numbers, which is countable, and therefore is a theoretical statement. Since we can only experimentally verify finite precision intervals, each verifiable statement will include infinitely many rational (and irrational) numbers. Therefore no verifiable statement is narrower than $\bar{\stmt}$ and therefore $\ver(\bar{stmt}) \equiv \impossibility$. But for the same reason no verifiable statement is incompatible with $\bar{stmt}$ and therefore $\fal(\bar{\stmt}) \equiv \impossibility$. Which means $\und(\bar{\stmt}) \equiv \certainty$. This means that the experimental test for $\stmt$ will never terminate either successfully or unsuccessfully. We call this type of statements undecidable as we will never be able to experimentally test anything about them. \begin{mathSection} \begin{defn}