I appreciate this post. It did, however, make me want to sit down and read Locke with more care. High praise indeed! Poor Whitehead! His later work has appealed to second-raters, so he ends up being associated with the intellectual backwaters of American philosophy.
This is a pity, because much of his work is first rate. There was originally supposed to be a fourth volume of PM, devoted to Geometry, and written by Whitehead alone. It never came out, but some of the work on the foundations of geometry that would have gone with more symbols! It influenced, among others, Tarski. Pingback: Why Check A Proof? You are commenting using your WordPress. You are commenting using your Google account.
You are commenting using your Twitter account. You are commenting using your Facebook account. Notify me of new comments via email. Notify me of new posts via email. You have to take the data of history seriously. Culture is part of the unholy trinity—culture, chaos, and cock-up—which roam through our versions of history, substituting for traditional theories of causation. The Renaissance Mathematicus.
Skip to content. Repeat after me! Alfred North Whitehead. Like this: Like Loading Readers today i. Readers wanting assistance are advised to consult the entry on the notation in Principia Mathematica. Even so, the book remains one of the great scientific documents of the twentieth century.
The system of propositional logic of PM, can be seen as a system of sentential logic consisting of a language, and rules of inference. PM contains the first presentation of symbolic logic that deals with propositional logic as a separate theory. In this section we will use A , B , etc as meta-linguistic variables for formulas. The formulas constructed from atomic propositions with the connectives are said to express elementary propositions to distinguish them from propositions involving quantifiers and propositional functions.
The propositional calculus is characterized by the fact that all its propositions have as hypothesis and as consequent the assertion of a material implication. That the system of propositional logic in PM was the result of an evolution of changes in choices of primitives is mirrored in the choice of theorems that are proved in the first chapters.
While most are proved because they will be used later in PM, some remain simply as remants of the earlier systems. In particular PM contains several theorems that were primitive propositions in earlier systems, though not used in what follows.
The notion of truth-functional semantics for propositional logic, using the familiar truth tables, and the notion of completeness of an axiom system, was not developed until soon after the publication of PM by Bernays There is no explicit statement of a rule of substitution in PM. The free variables in the propositional logic of PM may be interpreted as schematic letters, and so the system will require a rule of substitution of formulas.
In this article they are to be interpreted as real variables ranging over propositions, in which case instances would be derived by instantiation from generalizations over all propositions. The announcement in the Introduction that propositions are not necessary in what follows and so will be avoided suggests the schematic interpretation of the variables.
This interpretation of the letters as variables will also assist in the presentation of quantificational logic in PM below. As is standard for an axiomatic formulation of logic, a derivation of a formula of sentential logic in PM will consist of an instance of one of the six axioms, the result of a substitution in a preceding line, or the application of modus ponens to two preceding lines.
Theorems of PM will be proved in order, allowing the use of instances of preceding theorems as lines in later derivations. The resulting system is complete, in the sense that all and only truth-functionally valid sentences are derivable in the system.
This despite the seeming defects of the system by modern standards, including the redundancy of one of the axioms, the use of defined symbols in expressions to which the rules of inference apply, and the use of defined symbols in the axioms. Theorems are proved primarily as needed in later numbers, but some were axioms, or important theorems of earlier versions of propositional logic, going back to The Principles of Mathematics. Aside from historical interest in their actual choices, however, the system of PM can be viewed as based on any standard system of propositional logic.
The theory of types in the initial chapters of PM is ramified , so that within a given type, of propositions, or of functions of individuals, and functions of functions of individuals, there will be finer subdivisions.
The most prominent of these is the propositional Liar paradox created by the proposition that all propositions of a certain sort, say asserted by Epimenides, are false, when that very proposition is of that sort, that is the only proposition that Epimenides asserted. The solution in the ramified theory of types requires that a proposition about a sort of first level propositions, say that they are all false, will itself be of the next order. The paradoxes of the theory of sets are resolved by reducing assertions about sets to assertions about propositional functions.
The restriction that a function of one type cannot apply to a function of the same type is enough to block the paradoxes. In the Introduction to PM terminology is introduced for the two ways that variables may appear in formulas. The proper interpretation of higher-order variables in PM is the subject of contemporary dispute among scholars of PM. Landini and Linsky offer two rival accounts. Then the more distinctive notions of PM that depend on the theory of types can be explained.
The Axiom of Reducibility asserts that for an arbitrary function of any order there is an equivalent predicative function, that is, one true of exactly the same range of arguments. This will be explained below. Although PM does not single out first order logic from the whole ramified theory of types, the actual deductive apparatus on the page looks exactly like a system of first order logic, and the complications of the logic of higher types can be expressed with an additional apparatus of type indices.
In what follows we will use the system of r-types in Church for type indices, and the use of lambda operators for propositional functions. Note that there are two kinds of variables, but they are all assigned to an r-type. Individual variables behave as a special case of propositional function variables. The system of symbols for r -types and the assignment of r -types to variables for different entities individuals and functions is as follows:.
There are no predicate or individual names in this language. There are, however complex terms for propositional functions, defined together with formulas with the usual notion of bound and free variables :. We then can define the well formed formulas wffs and terms of quantificational logic as follows:.
The comprehension principle for a system of higher-order logic, or set theory, states which formulas express a property or set. The comprehension principle then is characterized by an infinite set of sentences of the form of:. Quine The offense comes from attributing orders r -types to propositional functions on the basis of the variables with which they are defined, but also to the functions themselves, as simply values of bound higher-order variables.
In response, the defender of type theory must say that any semantic intrepretation of the notion of propositional function will have to attribute to functions these distinctions that are marked in linguistic expressions of some of them, and in particular, the variables involved in their definition. This shows the extent to which the earlier theory is indeed a theory of propositions , not an account of a fragment of quantificational logic allowing open sentences containing free variables.
While of interest to scholars of PM, the upshot is the same for later uses of quantificational logic in PM. Again, the reader interested in what distinguishes the logicist project in PM can skip this section, although passing attention may be paid to the system of higher-order logic that is used, as based here on the ramified theory of types.
The extension to functions of more than one variable is obvious, and below, some applications will employ this extension. In what follows A is now an arbitrary possibly quantificational formula :. In part this is because of the application to an argument of lambda expressions for a propositional function, e. Some that are often used in later numbers are:. In other words, this section introduces the logic of quantification, in a way that is familiar to contemporary logic.
Consider the fundamental notion from the theory of real numbers of the least upper bound l. Consider the class of all real numbers whose square is less than or equal to 2, i. The resolution of this in the system of PM is to adopt an axiom which guarantees that any class defined in terms of another class will be of the same type. Thus impredicative definitions of classes are allowed, and do not introduce a class of a higher type. More precisely, the Axiom of Reducibility asserts that for any function of any number of arguments of an arbitrary level, there is an equivalent function of level 1, ie.
This notion of predicative functions is taken from the Introduction. See the accompanying entry on the notation in Principia Mathematica. It has seemed to some, beginning with Chwistek and continuing through Copi that the Axiom of Reducibility is technically faulty, leading to an inconsistency, or at least redundancy in the system of PM.
Ramsey early on argued that the supposed contradiction in fact demonstrated that certain predicative functions are indefinable. Church , confirms this assessment, and uses the presentation of r -types we describe here to show rigorously the limitations on what functions are definable in the system of PM.
In PM the notion of identity is defined following Leibniz as indiscernibility, namely indiscernible objects are identical. But since the axiom of reducibility guarantees that if there is any type of function on which x and y differ, they will differ on some predicative function, PM uses the following definition of identity:.
In other words, identicals are indiscernible. The given definition of identity only suffices if it is not possible that entities x and y which share all predicative properties, cannot be distinguished by some property of a higher order. In the appendix B to the second edition of PM, which was written by Russell, there is a technical discussion of the consequences of abandoning the axiom of reducibility. A faulty proof is proposed to show that the principle of Induction can be derived without using the axiom of reducibilty in a modified theory of types see Linsky The thesis that every class of reals with an upper bound has a real number as its least upper bound, discussed above, would not be provable.
The theory of definite descriptions is essential for this argument. The technical purpose, however, does indicate an important distinction between the logicism of Frege and Russell. Some logicians firmly in the tradition of mathematical logic do not find this to be an advance, but it does indicate a significant difference between the approaches of Frege and Russell see Linsky The latter is the reading on which it is not the case that there is one and only one present King of France and he is bald.
That may be true if there is not exactly one present King of France, as is actually the case, as France has no King. These expressions do not figure in theorems later in PM and only occasionally in the introductory material of some sections. This theorem is another indication of the way in which the philosophical basis of PM, with its propositional functions that are intensional is left behind as the mathematical content of PM is introduced with the definition of classes in the next sections.
The theory of sets classes in PM is based on a number of contextual definitions, similar in some ways to the theory of descriptions. The basic definition eliminates terms for classes from contexts in which they occur, just as the theory of definite descriptions eliminates descriptions occuring in the positions of terms:.
After these foundational sections, all the individual variables that appear in PM should be seen as ranging over classes, and, as will be explained below, the relation symbols are to be interpreted as ranging over relations in extension.
The paradox arises when one asks whether that class is a member of iteself or not. A function must be of a higher order than its arguments. The definitions of existential and universal quantification are simple. Because formulas with Greek variables look and behave the same as individual variables with respect to quantificational logic, it is possible to overlook the interaction of the theory of classes with the theory of types. Linsky argues that PM has no notation for classes of propositional functions to distinguish them from classes of classes, although one could be added.
This means that variables and terms for classes will obey the simple theory of types. It should be noted that every s -type is also an r -type, namely one that is hereditarily predicative. Thus it might seem that the expressions of the theory of classes are all simply a special case of formulas of the full system of the ramified theory of types.
This is the step comparable to the proof that a definite description is proper, i. It is widely thought that the system of PM offers a very different approach to the solution of the paradoxes than that of axiomatic set theory as formulated in the Zermelo-Fraenkel system ZF. This view has been forcefully expressed by Quine:. Whatever the inconveniences of type theory, contradictions such as [the Russell paradox] show clearly enough that the previous naive logic needs reforming.
None has the backing of common sense. Common sense is bankrupt, for it wound up in contradiction. At least hitherto only one solution which meets these two requirements [of avoiding the paradoxes while retaining mathematics and the theory of aggregates] has been found. It is the same intuition that underlies the hierarchy of types. Strictly as presented in PM, however, the no-classes theory differs significantly from ZF.
The sentences of the PM theory are expressed in the theory of types, as opposed to the first order theory of ZF. ZF and PM cannot simply be compared in terms of their theorems. Not only are there different axioms in the two theories, but the very languages in which they are expressed differ in logical power. As Quine remarks in his study of the logic of Whitehead and Russell, it would seem that after a certain point the body of PM makes use of extensional higher-order logic in a simple theory of types:.
In any case there are no specific attributes [propositional functions] that can be proved in Principia to be true of just the same things and yet to differ from one another. The theory of attributes receives no application, therefore, for which the theory of classes would not have served.
Once classes have been introduced, attributes are scarcely mentioned again in the course of the three volumes. Quine here hints at the view of PM that is widely shared among mathematical logicians, who see the ramified theory of types, with its accompanying Axiom or Reducibility, as a digression taking logic into a realm of obscure intensional notions, when instead logic, even if expressed in a theory of types, is extensional and is comparable to axiomatic set theory presented with a simple hierarchy of sets of individuals, sets of sets individuals, and so on.
It is certainly true that the the remainder of PM is devoted to the theory of individuals, classes, and relations in extension between those entities. Thus the ontology of these later portions is a hierarchy of predicative functions arranged in a simple theory of types. This has led one interpreter, Gregory Landini , to argue that only predicative functions are values of bound variables in PM.
The only bound variables in PM, he asserts, range over predicative functions. This is a strong version of a view that others such as Kanamori have expressed, going back to Ramsey , namely that the introduction of the Axiom of Reducibility has the effect of undoing the ramification of the theory of types, at least for a theory of classes, and so a higher-order logic used for the foundations of mathematics ought to have only a simple type structure.
In the summary of the later sections of PM that follows below, it will appear that in fact the symbolic development follows very closely that of PoM from ten years earlier.
To remind the reader of the change from talking of propositional functions to relations in extension, two further notational alterations are introduced.
The obvious limitation of this notation is that it is not readily extended to three place relations, adding a third variable, say z. The notions of the subset relation and the intersection and union of sets are defined in PM exactly as they are now albeit with different terminology.
The complement of a set of a given type is the set of all entities of that type that are not in the set. There is no class of all classes of whatever type. This is in common with axiomatic set theory which holds that there is no set of all sets. If there is a binary relation which has a unique second argument for each first argument, i.
The definition of a monadic functional term then is:. The diligent reader will find that this presentation does not follow PM exactly. The practice of reading the argument of a relational function as the x and the value as the y is so well established that we have taken a liberty with the actual definitions in PM. A series of notions are defined in a way quite familiar to the modern treatment of relations as sets of n -tuples:. The notions of the domain , range , and field of a relation are also given a contemporary definition and so also the notions of the domain , range and field of a function.
Note that it is possible that a relation can have its domain in one type and range in another. This adds complications in the theory of cardinal numbers when a relation of similarity equinumerousity holds between classes of different types. In his survey of PM, Quine complains that this last pages of Part I is occupied with proving theorems relating redundant definitions of the same notions.
Thus PM defines the notion of domain and range and then introduces notions that again define the same classes, which are proved to be equivalent.
PM, In contemporary logic with the notation of set theory used above, there is no need for a special symbol for this notion, as it is written as:. So the cardinal number 1 is the class of all singletons. There will be a different number 1 for each type of x.
Frege, by contrast, defines the natural number 1 as the extension of a certain concept, namely being identical with the number 0, which itself is the extension of the empty concept of not being self identical. This construction is named the von Neumann ordinals. Similarly, the number 2 is the class of all pairs, rather than a particular pair. In the type theory of PM there will be distinct couples for the types of y and x.
Even with homogenous pairs there will be distinct classes of pairs for each type, and thus a different number 2 for each type. The same notion applies to relations. It is a relation in extension, which is the analogue of a property in extension or class. A relation in extension has a distinction between the first and second elements due to the order of the defining relation.
The closest in contemporary language would be:. But why start from logic? I think Russell just assumed that logic was the most fundamental possible thing—the ultimate incontrovertible representation for all formal processes. Traditional mathematical constructs—like numbers and space—he imagined were associated with the particulars of our world.
In my own work leading up to A New Kind of Science , I started by studying the natural world, yet found myself increasingly being led to generalize beyond traditional mathematical constructs. But I did not wind up with logic. Instead, I began to consider all possible kinds of rules—or as I have tended to describe it making use of modern experience , the computational universe of all possible programs.
Some of these programs describe parts of the natural world. Some give us interesting fodder for technology. And some correspond to traditional formal systems like logic and mathematics. One thing to do is to look at the space of all possible axiom systems. There are some technical issues about modern equational systems compared to implicational systems of the kind considered in Principia Mathematica. But the essential result is that dotted around the space of all possible axiom systems are the particular axiom systems that have historically arisen in the development of mathematics and related fields.
In Principia Mathematica , Russell and Whitehead originally defined logic using a fairly complicated traditional set of axioms. In the second edition of the book, they made a point of noting that by writing everything in terms of Nand Sheffer stroke rather than And, Or and Not, it is possible to use a much simpler axiom system.
In , by doing a search of the space of possible axiom systems, I was able to find the very simplest equational axiom system for standard propositional logic: just the single axiom a. And from this result, we can tell where logic lies in the space of possible formal systems: in a natural enumeration of axiom systems in order of size, it is about the 50,th formal system that one would encounter.
A few other traditional areas of mathematics—like group theory—occur in a comparable place. But most require much larger axiom systems. And in the end the picture seems very different from the one Russell and Whitehead imagined. It is not that logic—as conceived by human thought—is at the root of everything. Instead, there are a multitude of possible formal systems, some picked by the natural world, some picked by historical developments in mathematics, but most out there uninvestigated.
And indeed the very heft of the book gave immediate support to this idea, and gave such credibility to logic and to Russell that Russell was able to spend much of the rest of his long life confidently presenting logic as a successful way to address moral, social and political issues. Of course, in Kurt Godel showed that no finite system—logic or anything else—can be used to derive all of mathematics. And indeed the very title of his paper refers to the incompleteness of none other than the formal system of Principia Mathematica.
So can one say that the idea of logic somehow underlying mathematics is wrong? At a conceptual level, I think so. But in a strange twist of history, logic is currently precisely what is actually used to implement mathematics.
For inside all current computers are circuits consisting of millions of logic gates—each typically performing a Nand operation. And so, for example, when Mathematica runs on a computer, and implements the operations of mathematics, it does so precisely by marshalling the logic operations in the hardware of the computer.
To be clear, the logic implemented by computers is basic, propositional, logic—not the more elaborate predicate logic, combined with set theory, that Principia Mathematica ultimately uses. We know from computational universality—and more precisely from the Principle of Computational Equivalence —that things do not have to work this way, and that there are many very different bases for computation that could be used.
And indeed, as computers move to a molecular scale, standard logic will most likely no longer be the most convenient basis to use. I suspect it actually has quite a bit to do with none other than Principia Mathematica. For historically Principia Mathematica did much to promote the importance and primacy of logic, and the glow that it left is in many ways still with us today.
It is just that we now understand that logic is just one possible basis for what we can do—not the only conceivable one. I think this is a matter of definition, but in any case, what has become clear for mathematics is that it is vastly more important to compute answers than merely to state truths.
A hundred years after Principia Mathematica there is still much that we do not understand even about basic questions in the foundations of mathematics. And it is humbling to wonder what progress could be made over the next hundred years. When we look at Principia Mathematica , it emphasizes exhibiting particular truths of mathematics that its authors derived.
But today Mathematica in effect every day automatically delivers millions of truths of mathematics, made to order for a multitude of particular purposes. Yet it is still the case that it operates with just a few formal systems that happen to have been studied in mathematics or elsewhere. And even in A New Kind of Science , I concentrated on particular programs or systems that for one reason or another I thought were interesting. In the future, however, I suspect that there will be another level of automation.
Probably it will take much less than a hundred years, but in time it will become commonplace not just to make computations to order, but to make to order the very systems on which those computations are based—in effect in the blink of an eye inventing and developing something like a whole Principia Mathematica to respond to some particular purpose.
Posted in: Historical Perspectives , Mathematics , Philosophy. Please enter your comment at least 5 characters. Please enter a valid email address. Thank you, thank you. This is an amazing post. A wonderful alternative to football after thanksgiving dinner. Please keep up the great blog posts.
Given equivalence in all its forms, there has never been a reason one could not construct something similar about the exact same subject — or any subject — using any equivalent set of axioms. They could be a different axiom, a different symbol and meaning-set, or something as abstracted — and otherwise concrete — as levers, pulleys, rubber-bands, rope and bearings.
What type of problem might yield to this system? Might our very notions of computability be challenged? Thanks for your message. You touch upon an enthralling concept, the concept of truth in mathematics.
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