on 08-Dec-2023 (Fri)

由于该公式在\(x\)为复数时仍然成立，所以也有人将这一更通用的版本称为欧拉公式：\(e^θ = \cosh θ + \sinh θ\)

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In spite of the polar opposition between the two schools, they depend upon acommon premise. According to both sys- tems of philosophy, reflective thought, thinking that involves Inference and judgment, is not originative. It has its test in antecedent reality as that is disclosed in some non-reflective immediate knowledge. Its validity depends upon the possi- bility of checking its conclusions by identification with the terms of such prior immediate knowledge. The controversy between the schools is simply as to the organ and nature of previous direct knowledge. To both schools, reflection, thought involving inference, is reproductive jthe "proof" of its results is found in comparison with what is known without any infer- ence. In traditional empiricism the test is found in sensory im- pressions. For objective idealism, reflective inquiry is valid only as it reproduces the work previously effected by constitutive thought.

the direction of experi- ment by ideas, the fact that experiment is not random, aimless action, but always includes, along with groping and relatively blind doing, an element of deliberate foresight and intent, which determines that one operation rather than another be tried.

Let us suppose, for the time being, that all that we can know about ideas is derived from the way in which they figure in the reflective inquiries of science. What con- ception of their nature and office shall we then be led to form?

In the old text-books mass was defined as 'quantity of matter' jbut when it came to an actual determination of mass, an experimental method was prescribed which had no bearing on this definition." f

oncepts are recognized by means of the experimental operations by which they are determined; that is, operations define and test the validity of the meanings by which we state natural happen- ings.

The principle is anticipated in Peirce's essay on How to Make Our Ideas Clear

Peirce states that the sole meaning of the idea of an object consists of the consequences which result when the object is acted upon in aparticular way. The principle is one element in the pragma- tism of James. The idea is also akin to the "instrumental" theory of concep- tions, according to which they are intellectual instruments for directing our activities in relation to existence.

what has been lacking throughout the history of thought, agenuinely experimental empiricism.

From the standpoint of the operational definition and tests of ideas, ideas having an empirical origin and status. But it is that of acts performed, acts in the literal and existential sense of the word, deeds done, not reception of sensations forced on us from without. Sensory qualities are important. But they are intellectually significant only as consequences of acts inten- tionally performed.

To suppose that its cognitive value can be eked out or supplied by associating it with other sensory qualities of the same nature as itself, is like supposing that by putting apile of sand in the eye we can get rid of the irritation caused by asingle grain. To suppose, on the other hand, that we must appeal to asynthetic activity of an independent thought to give the quality meaning in and for knowledge, is like suppos- ing that by thinking in our heads we can convert apile of bricks into abuilding. Thinking, carried on inside the head, can make some headway in forming the f)lan of abuilding. But it takes actual operations to which the plan, as the fruit of thought, gives instrumental guidance to make abuilding out of separate bricks, or to transform an isolated sensory quality into asignificant clew to knowledge of nature.

Sensory qualities experienced through vision have their cognitive status and office, not (as sensational empiricism holds) in and of themselves in isolation, or as merely forced upon attention, but because they are the consequences of definite and intentionally performed operations. Only in connection with the intent, or idea, of these operations do they amount to anything, either as disclosing any fact or giving test and proof of any theory. The rationalist school was right in as far as it insisted that sensory qualities are significant for knowledge only when connected by means of ideas. But they were wrong in locating the connecting ideas in intellect apart from experi- ence. Connection is instituted through operations which define ideas, and operations are as much matters of experience as are sensory qualities.

an empirical theory of ideas free from the burdens imposed alike by sensationalism and apriori ra- tionalism. This accomplishment is, Imake bold to say, one of three or four outstanding feats of intellectual history. For it emancipates us from the supposed need of always harking back to what has already been given, something had by alleged direct or immediate knowledge in the past, for the test of the value of ideas. Adefinition of the nature of ideas in terms of operations to be performed and the test of the validity of the ideas by the consequences of these operations establishes connectivity within concrete experience. At the same time, by emancipation of thinking from the necessity of testing its con- clusions solely by reference to antecedent existence it makes clear the originative possibilities of thinking.

How could the unity of anything be secure unless there was some- thing persistent and unchanging behind all change? Without such fixed indissoluble unities, no final certainty was possible. Everything was put in peril of dissolution. These metaphysi- cal fears rather than any experimental evidence determined the nature of the fundamental assumptions of Newton regarding atoms.

t was logically inevitable that as science proceeded on its experimental path it would sooner or later become clear that all conceptions, all intellectual descriptions, must be formu- lated in terms of operations, actual or imaginatively possible. There are no conceivable ways in which the existence of ulti- mate unchangeable substances which interact without under- going change in themselves can be reached by means of experimental operations. Hence they have no empirical, no ex- perimental standing jthey are pure dialectic inventions:

the minds of men, including physical inquirers, were still possessed by the old notion that reality in order to be solid and firm must con- sist of those fixed immutable things which philosophy calls substances. Changes could be known only if they could be somehow reduced to recombinations of original unchanging things. For these alone can be objects of certainty the chang- ing is as such the uncertain and only the certain and exact is knowledge. Thus apopular metaphysics, given rational for- mulation by the Greeks, and taken over into the intellectual tradition of the western world, controlled at first the inter- pretations placed upon the procedures and conclusions of ex- perimental knowing.

The fact that Newton adopted the Democritean rather than the Aristotelian conception of substance is of course of immense importance scientifically. But philosophically speaking it is of slight import compared with the fact that he followed the supposed necessities of dialectic reasoning rather than the lead of experienced subject-matter

in spite of starting upon the experimental and mathematical path, Newtonian science kept the idea that atoms are characterized by eternal properties or qualities, that is by essences. Substances are "solid, hard, massy, impenetrable, moveable particles." Their essence is precisely these unchange- able, fixed qualities of solidity, mass, motion, inertia. It thus appears that Newton retained apart of the qualita- tive equipment of the objects of Greek science, in spite of their irrelevance to both mathematics and experiment.

It is at this point that the recent recognition that the con- ceptions by which we think scientific objects are derived neither from sense nor from apriori conceptions has its logical and philosophical force. Sense qualities, as we saw in the previous chapter, are something to be known, they are challenges to knowing, setting problems for investigation. Our scientific knowledge is something about them, resolving the problems

Inquiry proceeds by reflection, fcy thinking; but not, most decidedly, by thinking as conceived in the old tradition, as something cooped up within "mind." For ex- perimental inquiry or thinking signifies directed activityydoing something which varies the conditions under which objects are observed and directly had and by instituting new arrange- ments among them. Things perceived suggest to us (originally just evoke or stimulate) certain ways of responding to them, of treating them.

The first effect of experimental analysis is, as we saw, to reduce objects directly experienced to data. This resolution is required because the objects in their first mode of experience are perplexing, ob- scure, fragmentary} in some way they fail to answer aneed. Given data which locate the nature of the problem, there is evoked athought of an operation which if put into execution may eventuate in asituation in which the trouble or doubt which evoked inquiry will be resolved. If one were to trace the history of science far enough, one would reach atime in which the acts which dealt with atroublesome situation would be organic responses of astructural type together with afew acquired habits. The most elaborate technique of present in- quiry in the laboratory is an extension and refinement of these simple original operations.

Often the invention of atool suggested operations not in mind when it was in- vented and thus carried the perfecting of operations still fur- ther. There is thus no apriori test or rule for the determina- tion of the operations which define ideas. They are themselves experimentally developed in the course of actual inquiries. They originated in what men naturally do and are tested and improved in the course of doing.

Consequences that successfully solve the prob- lems set by the conditions which give rise to the need of action supply the basis by means of which acts, originally "naturally5'performed, become the operations of the art of scientific experimentation.

there is one common character of all such scientific operations which it is necessary to note. They are such as disclose relationships. A simple case is the operation by which length is defined by one object placed end upon end upon another object so many times. This type of operation, repeated under conditions them- selves defined by specified operations, not merely fixes the relation of two things to each other called their length, but defines ageneralized concept of length. This conception in connection with other operations, such as those which define mass and time, become instruments by means of which a multitude of relations between bodies can be established. Thus the conceptions which define units of measurement of space, time and motion become the intellectual instrumentalities by which all sorts of things with no qualitative similarity with one another can be compared and brought within the same system. To the original gross experience of things there is superadded another type of experience, the product of deliber- ate art, of which relations rather than qualities are the signifi- cant subject-matter. These connections are as much experi- enced as are the qualitatively diverse and irreducible objects of original natural experiences.

ualities present themselves as just what they are, stati- cally demarcated from one another. Moreover, they rarely change, when left to themselves, in such ways as to indicate the interactions or relations upon which their occurrence de- pends. No one ever observed the production of the thing having the properties of water, nor the mode of generation of aflash of lightning. In sensory perception the qualities are either too static or too abruptly discrete to manifest the specific connections that are involved in their coming into existence. Intentional variation of conditions gives an idea of these connections.

or along time the definitions were supposed to be made not in terms of relations but through certain properties of antecedent things. The space, time and motion of physics were treated as inherent properties of Be- ing, instead of as abstracted relations. In fact, two phases of inquiry accompany each other and correspond to each other. In one of these phases, everything in qualitative objects ex- cept their happening is ignored, attention being paid to quali- ties only as signs of the nature of the particular happening in question: that is, objects are treated as events. In the other phase, the aim of inquiry is to correlate events with one an- other. Scientific conceptions of space, time and motion consti- tute the generalized system of these correlations of events. Thus they are doubly dependent upon operations of experi- mental art: upon those which treat qualitative objects as events, and upon those which connect events thus determined with one another.

Cer- tain properties regarded by Newton as inherent in substances and essential to them, in independence of connectivity, were indeed speedily seen to be relations. This conversion hap- pened first as to hardness and impenetrability, which were seen to be reducible to mass. Vis inertiae was ameasure of mass. By careful thinkers "force" was treated as ameasure of acceleration and so aname for arelation, not as an inherent property of an isolated thing by virtue of which one thing could compel another to change.

o say this is not to disparage the scientific importance of the discovery that mass varies with velocity and of the result of the Michelson-Morley experiment on the velocity of light. Such discoveries were doubtless necessary in order to force recognition of the operational or relational character of scien- tific conceptions. And yet, logically, the way in which space, time and motion, with their various functions, appear in mathe- matical equations and are translated into equivalent formula- tions with respect to one another something which is impos- sible for qualities as such indicates that arelational treat- ment had always been involved. But the imagination of men had become used to ideas framed on the pattern of large masses and relatively slow velocities. It required observation of changes of high velocity, as of light over great distances, and of minute changes occurring at infinitesimal distances to emancipate imagination from its acquired habitudes.

The dis- covery that mass varies with velocity did away with the pos- sibility of continuing to suppose that mass is the defining characteristic of things in isolation from one another such isolation being the sole condition under which mass could be regarded as immutable or fixed.

With the surrender of unchangeable substances having properties fixed in isolation and unaffected by inter- actions, must go the notion that certainty is attained by attach- ment to fixed objects with fixed characters. For not only are no such objects found to exist, but the very nature of ex- perimental method, najmely, definition by operations that are interactions, implies that such things are not capable of being known. Henceforth the quest for certainty becomes the search for methods of control jthat is, reguktion of conditions of change with respect to their consequences.

Theoretical certitude is assimilated to practical certainty j to securityytrustworthiness of instrumental operations. "Real" things may be as transitory as you please or as lasting in time as you please; these are specific differences like that between aflash of lightning and the history of amountain range. In any case they are for knowledge "events" not substances. What knowledge is interested in is the correlation* among these changes or events which means in effect that the event called the mountain range must be placed within asystem consisting of avast multitude of included events. When these correlations are discovered, the possibility of control is in our hands. Scientific objects as statements of these inter-relations are instrumentalities of control. They are objects of the thought of reality, not disclosures of immanent properties of real sub- stances. They are in particular the thought of reality from a particular point of view: the most highly generalized view of nature as asystem of interconnected changes.

That heat is amode of motion does not signify that heat and cold as qualitatively experienced are "unreal," but that the qualitative experience can be treated as an event measured in terms of units of velocity of movement, involving units of position and time, so that it can be connected with other events or changes simi- larly formulated. The test of the validity of any particular intellectual conception, measurement or enumeration is func- tional, its use in making possible the institution of interactions which yield results in control of actual experiences of observed objects.

Prisoners in jails are often given numbers and are "known" by the numbers assigned. It has not yet occurred to any one that these numbers are the real prisoners, and that there is always aduplicate real object jone a number, and the other aflesh and blood person, and that these two editions of reality have to be reconciled. It is true that the numbers which constitute by means of measurements the object of scientific thought are not assigned so arbitrarily as those of prisoners, but there is no difference in philosophical principle.

Put positively, the physical ob- ject, as scientifically defined, is not aduplicated real object, but is astatement, as numerically definite as is possible, of the relations between sets of changes the qualitative object sus- tains with changes in other things ideally of all things with which interaction might under any circumstances take place.

Since these correlations are what physical inquiry does know, it is fair to conclude that they are what it intends or means to know: on analogy with the legal maxim that any reasonable person intends the reasonably probable consequences of what he does. We come back again to the frequently re- peated statement that the problem which has given so much trouble to modern philosophy that of reconciling the reality of the physical object of science with the richly qualitative object of ordinary experience, is afactitious one. All that is required in order to apprehend that scientific knowledge as a mode of active operation is apotential ally of the modes of action which sustain values in existence, is to surrender the traditional notion that knowledge is possession of the inner nature of things and is the only way in which they may be experienced as they "really" are.

In itself, the object is just what it is experienced as being, hard, heavy, sweet, sonorous, agreeable or tedious and so on. But in being "there" these traits are effects, not causes. They cannot as such be used as means, and when they are set up as ends in view, we are at aloss how to secure them. For just as qualities there are no constant and definite relations which can be ascertained between them and other things.

ahierarchy of species and genera to see the great gain that has been effected. It is the very nature of fixed kinds to be as exclusive with respect to those of adifferent order as it is to be inclusive with respect to those which fall within the class. Instead of athoroughfare from one order to another, there was asign: No passage. The work of emancipation which was initiated by experimentation, setting objects free from limita- tion by old habits and customs, reducing them to acollection of data forming aproblem for inquiry, is perfected by the method of conceiving and defining objects through operations which have as their consequence accurate metric statements of changes correlated with changes going on elsewhere.

The resolution of objects and nature as awhole into facts stated exclusively in terms of quantities which may be handled in calculation, such as saying that red is such anumber of changes while green is another, seems strange and puzzling only when we fail to appreciate what it signifies. In reality, it is adeclaration that this is the effective way to think things j the effective mode in which to frame ideas of them, to formu- late their meanings. The procedure does not vary in principle from that by which it is stated that an article is worth so many dollars and cents. The latter statement does not say that the article is literally or in its ultimate "reality" so many dollars and cents jit says that for purpose of exchange that is the way to think of it, to judge it. It has many other mean- ings and these others are usually more important inherently. But with respect to trade, it is what it is worth, what it will sell for, and the price value put upon it expresses the relation it bears to other things in exchange. The advantage in stating its worth in terms of an abstract measure of exchange such as money, instead of in terms of the amount of corn, potatoes or some other special thing it will exchange for, is that the latter method is restricted and the former generalized.

The formulation of ideas of experienced objects in terms of measured quantities, as these are established by an inten- tional art or technique, does not say that this is the way they must be thought, the only valid way of thinking them. It states that for the purpose of generalized, indefinitely exten- sive translation from one idea to another, this is the way to think them. The statement is like any other statement about instruments, such as that so-and-so is the best way of sending anumber of telegraphic dispatches simultaneously. As far as it is actually the best instrumentality, the statement is correct. It has to be proved by working better than any other agency jit is in process of continuous revision and improvement. For pur- poses except that of general and extensive translation of one conception into another, it does not follow that the "scientific" way is the best way of thinking an affair. The nearer we come to an action that is to have an individualized unique object of experience for its conclusion, the less do we think the things in question in these exclusively metric terms. The physi- cian in practice will not think in terms as general and abstract as those of the physiologist in the laboratory, nor the engineer in the field in those as free from special application as will the physicist in his work-shop. There are many ways of think- ing things in relation to one another jthey are, as conceptions, instruments. The value of an instrument depends upon what is to be done with it. The fine scale micrometer which is indis- pensable in the successful performance of one operation would be ahindrance in some other needed actj and awatch spring is useless to give elasticity to amattress.

these scientific conceptions, like other instruments, are hand-made by man in pursuit of realization of acertain interest that of the maxi- mum convertibility of every object of thought into any and every other. It is awonderful ideal jthe ingenuity which man has shown in devising means of realizing the interest is even more marvelous. But these ways of thinking are no more rivals of or substitutes for objects as directly perceived and enjoyed than the power-loom, which is amore effective in- strument for weaving cloth than was the old hand-loom, is a substitute and rival for cloth. The man who is disappointed and tragic because he cannot wear aloom is in reality no more ridiculous than are the persons who feel troubled because the objects of scientific conception of natural things have not the same uses and values as the things of direct experience.

The disconcerting aspect of the situation resides in the difficulty with which mankind throws off beliefs that have be- come habitual. The test of ideas, of thinking generally, is found in the consequences of the acts to which the ideas lead, that is in the new arrangements of things which are brought into existence. Such is the unequivocal evidence as to the worth of ideas which is derived from observing their position and role in experimental knowing. But tradition makes the tests of ideas to be their agreement with some antecedent state of things. This change of outlook and standard from what pre- cedes to what comes after, from the retrospective to the pros- pective, from antecedents to consequences, is extremely hard to accomplish. Hence when the physical sciences describe ob- jects and the world as being such and such, it is thought that

The business of thought is not to conform to or reproduce the characters already possessed by objects but to judge them as potentialities of what they become through an indicated operation.

Idealistic philosophies have not been wrong in attaching vast importance and power to ideas. But in isolating their function and their test from action, they failed to grasp the point and place where ideas have aconstructive office. Agenuine idealism and one com- patible with science will emerge as soon as philosophy accepts the teaching of science that ideas are statements not of what is or has been but of acts to be performed. For then mankind will learn that, intellectually (that is, save for the esthetic enjoy- ment they afford, which is of course atrue value), ideas are worthless except as they pass into actions which rearrange and reconstruct in some way, be it little or large, the world in which we live. To magnify thought and ideas for their own sake apart from what they do (except, once more, estheti- cally) is to refuse to learn the lesson of the most authentic kind of knowledge the experimental and it is to reject the idealism which involves responsibility. To praise thinking above action because there is so much ill-considered action in the world is to help maintain the kind of aworld in which action occurs for narrow and transient purposes.

React DOM Hooks - Hooks for web applications which run in the browser DOM environment.

React DOM Components - React supports all of the browser built-in HTML and SVG components.

React DOM APIs - The react-dom package contains methods supported only in web applications.

Client APIs - The react-dom/client APIs let you render React components on the client (in the browser).

Server APIs - The react-dom/server APIs let you render React components to HTML on the server.

React-dom contains features that are only supported for web applications (which run in the browser DOM environment).

go tool

Go programs are organized into packages. A package is a collection of source files in the same directory that are compiled together. Functions, types, variables, and constants defined in one source file are visible to all other source files within the same package.

A repository contains one or more modules. A module is a collection of related Go packages that are released together. A Go repository typically contains only one module, located at the root of the repository. A file named go.mod there declares the module path : the import path prefix for all packages within the module. The module contains the packages in the directory containing its go.mod file as well as subdirectories of that directory, up to the next subdirectory containing another go.mod file (if any).

Note that you don't need to publish your code to a remote repository before you can build it. A module can be defined locally without belonging to a repository. However, it's a good habit to organize your code as if you will publish it someday.

Each module's path not only serves as an import path prefix for its packages, but also indicates where the go command should look to download it. For example, in order to download the module golang.org/x/tools, the go command would consult the repository indicated by https://golang.org/x/tools (described more here).

An import path is a string used to import a package. A package's import path is its module path joined with its subdirectory within the module. For example, the module github.com/google/go-cmp contains a package in the directory cmp/. That package's import path is github.com/google/go-cmp/cmp. Packages in the standard library do not have a module path prefix.

To compile and run a simple program, first choose a module path (we'll use example/user/hello) and create a go.mod file that declares it:

$ mkdir hello # Alternatively, clone it if it already exists in version control.
$ cd hello
$ go mod init example/user/hello
go: creating new go.mod: module example/user/hello
$ cat go.mod
module example/user/hello go 1.16
$

The first statement in a Go source file must be package name . Executable commands must always use package main.

Now you can build and install that program with the go tool:

$ go install example/user/hello
$

This command builds the hello command, producing an executable binary. It then installs that binary as $HOME/go/bin/hello (or, under Windows, %USERPROFILE%\go\bin\hello.exe).

The install directory is controlled by the GOPATH and GOBIN environment variables. If GOBIN is set, binaries are installed to that directory. If GOPATH is set, binaries are installed to the bin subdirectory of the first directory in the GOPATH list. Otherwise, binaries are installed to the bin subdirectory of the default GOPATH ($HOME/go or %USERPROFILE%\go).

You can use the go env command to portably set the default value for an environment variable for future go commands:

$ go env -w GOBIN=/somewhere/else/bin
$

To unset a variable previously set by go env -w, use go env -u:

$ go env -u GOBIN
$

Commands like go install apply within the context of the module containing the current working directory. If the working directory is not within the example/user/hello module, go install may fail.

For convenience, go commands accept paths relative to the working directory, and default to the package in the current working directory if no other path is given. So in our working directory, the following commands are all equivalent:

$ go install example/user/hello

$ go install .

$ go install

Next, let's run the program to ensure it works. For added convenience, we'll add the install directory to our PATH to make running binaries easy:

# Windows users should consult https://github.com/golang/go/wiki/SettingGOPATH
# for setting %PATH%.
$ export PATH=$PATH:$(dirname $(go list -f '{{.Target}}' .))
$ hello
Hello, world.
$

React Hooks - Use different React features from your components.

React Components - Documents built-in components that you can use in your JSX.

APIs - APIs that are useful for defining components.

React Directives - Provide instructions to bundlers compatible with React Server Components.

React Hooks - Use different React features from your components.

React Components - Documents built-in components that you can use in your JSX.

React APIs - APIs that are useful for defining components.

React Directives - Provide instructions to bundlers compatible with React Server Components.