PROBLEM ANALYSIS FORMULA INDETERMINATE SCIENCE-ANALYSIS DETERMINATE GEORGE ASHTON BLACK, PH.a ,. J .J 'tjt.^ CopightN" i^i Q> CDFXRIGRT DEPOSIT. PROBLEM SCIENCE=ANALYSIS FORMULA INDETERMINATE SCIENCE=ANALYSIS DETERMINATE BY GEORGE ASHTON BLACK, Ph.D. Man gewinnt dadurcli schon sehr viel, ■wenn man eine Menge von UntersTichungen \inter die Formel einer einzigen Aufgabe bringen kann. Denn dadurcli erleich- tert man sich nicht allein selbst sein eigenes Geschaft, indem man es sicb genau bestimmt, sondem aueb jedem anderen, der es priifen will, das Urtheil, ob wir unserem Vorhaben ein Geniige gethan haben oder nicbt. — Kant. NEW YORK FEINTED FOR THE AUTHOR 1916 Copyright, 1916, by George Ashton BijACk Published September, 1916 if ■ SEP 23 1916 THE DE VINNE PRESS C!,A437787 AXIOM If the analytical expression to be read equals^ is abstracted from any application of it, for instance in logic the classic application A=A and is considered by itself as a single principle of cognition, of some kind, then the mathematical construction a single principle f plane surface of cognition, of \ straight line — some kind [ two equal jiarallel straiglit lines = is found to determine only generally the singularity, in re- lation to the kind, of every single principle of cognition that was ever placed within the whole of the considered principle in the course of constructing it by degrees according to a constant rule; and this only general determination of a singular state of things is in turn found to prescribe the complete determination of the same. NOTE Increasing diversity in the application of a constant rule is the general but sufficient mark of deductive sequence in general. Invariably in this treatise successive = in the order of deduction successive fii'st=in the order of deduction first POSTULATE The mathematical construction {axiom negation 1 postulate — reality \ quality- definition = limitation J can be recognized as necessary and sufficient indifferently either to make the complete determination prescribed in axiom a thing done, or to determine and resolve in deduc- tive sequence aU. cases of the singular general problem first cognition of any kind NOTE The same three principles of cognition in the same order, namely plane surface straight line — two equal parallel straight lines = 3 are also necessary and sufficient to construct mathemati- cally Kant's classic metaphysical theory whether of the dif- ferent successive steps in the process of making any object by degrees completely known according to a constant rule, or of what constitutes perfection in any cognition referred to any object. Witness {intuition synthesis 1 concept — analysis [ method idea = dialectic J where the presence of two equivalent formulations of the theory is explained by the fact that Kant distinguishes the same cognitions, regarded from two different points of view, on the one hand as determinations of a known object, on the other hand as acts of a knowing subject. Use of classic analytical expressions in a new sense which can be gathered from the new context with mathematical certainty and precision has been made, not pointed out, hitherto; and will continue to be made, not pointed out, in the sequel. But in this place it is convenient to call atten- tion, once for all, to the use in question, by pointing to the example of it contained in the rest of this paragraph. The two equivalent formulations, when posited in one to one correspondence, but not yet mathematically constructed (intuition synthesis ] concept analysis - !■ method idea dialectic J perfectly differentiate the whole of possible cognition ac- cording to a constant rule, that is logically divide it into a series of empty compartments. The integration of the re- sulting differential equation, that is the filling of the said compartments by the only cognitions necessary and suf- ficient to fiJl them {intuition synthesis 1 concept — analysis [method idea = dialectic J first reduces the formulated theory to practice; first con- structs it mathematically; first makes, according to that theory, the unknown cognitive function in the sense of 4 required cognition of any kind according to a constant rule, by degrees completely known. The construction of the two equivalent formulations of the theory raises a certain problem. For the resulting series in terms of mathematics plan^ surface straight line — two equal parallel straight lines = and the two equivalent formulations in terms of meta- physics {intuition synthesis 1 concept analysis > method idea dialectic J differ as intuition and concept in respect of the same object, namely the theory. Therefore a corresponding idea remains to be invented or discovered. I find it in the necessary reference of the terms of the series to the comparison and distinction of them in respect of any singularity of any kind remarkable in any one of them. The resulting idea or dialectic negation 1 [If ^^^^ reality [ quality = -I — I = first ] postulate limitation J [ = J I definition supplies all that was wanting to the perfection of the first complete knowledge of the theory according to the theory itself as a constant rule. In the case of the notation intuition synthesis whether the indicated compartment is to be regarded as empty or as filled cannot be told on inspection of only that notation. If the two equivalent formulations of Kant's theory are merely posited in apposition to each other in a one to one correspondence constituting the differential equation {intuition synthesis 1 concept analysis [method idea dialectic J then all the indicated compartments are empty and require 5 to be filled ; that is the theory requires to be reduced to practice ; that is the equation requires to be constructed = integrated = satisfied When all are filled by the only cognitions necessary and sufficient to fill them {intuition synthesis 1 concept — analysis [method idea = dialectic J then the first compartment is in semblance still empty, but is in truth now filled by the cognition termed a plane in geometry. What is here in question is a certain new apph- cation of the classic mathematical principle of position ac- cording to which the same notation in different places has different meanings precisely determined by the context. For example, whether a straight hue is to be regarded as the analysis of a previously undivided plane into two equal parts that are connected in the line as a concept, or as the syn- thesis of the first part of the reproduction by degrees ac- cording to a constant rule of a plane equilateral triangle, of which part the hne is an intuition, depends in this treatise on the position of the Hne as immediately successive in the first case to the undivided plane, in the second case to the whole of the triangle. Another new application has already appeared in the case of the notation and stni another will presently appear in the case of the notation Science = Analysis In the most remarkable of these new applications the iden- tical notation in question is only a httle clear space some- how sufficiently indicated on the page. Two different mean- ings of it, as it appears when one and the same notation containing it intuition synthesis is met with in different places, have already been pointed out. Other different meanings of it in other different places remain to be pointed out in the sequel. 6 DEFINITION Science = cognition necessary and sufficient to re- solve all cases of a general problem in deductive sequence = Analysis. ^ PROBLEM Science = Analysis FORMULA Indeterminate Science = Analysis Determinate WORK Science = Analysis Science | predicate | Analysis «^i-ee{-t7^,f^^^^^^^^^^ NOTE Formula is a convenient notation of the logical division necessary and sufficient to determine the case of problem universally through different cases in deductive sequence. The case in which problem is required to be indeterminate is the singular case. The case in which problem is required to be determinate is the general case. The first moment of work is the only solution of the singular case. In semblance it is only problem itself. In truth it is no determination of problem in reference to the necessary correlate of any subject of discussion, namely, any predicate. The second moment of work, as any deter- mination of problem in reference to any predicate, is the first of aU possible solutions of the general case. The third moment of work, as the sequence of negative and affirmative determination of problem in reference to any predicate, is the general solution to the form of which all solutions of the general case that are different and successive in reference to the first can be reduced. The sequence of second and third moments is necessary and sufficient to solve the 7 general case universally. The whole of work in giving the only solution of the singular case, and the only necessary and universal solution of the general case, gives the required resolution of all cases. PROBLEM Science | not any predicate | Analysis NOTE This negative problem is resolved only through the absence of formula and work in the case of it. According to definition it is not resolved through connecting with the word not the predicates resulting from the resolution of the next successive problem. PROBLEM Science | any predicate | Analysis FORMULA ^-- { any sn'jl^sl^f ptdicate } ^^^^^ * WORK S-^--«{toc"on}^-^ly^i« according to classic definition of abstract in logic, of func- tion in mathematics. NOTE In logic classic distinction between abstract and applied so defines abstract that it cannot be any successive predicate, and can only be the first predicate, of which any successive predicate is some apphcation. In mathematics classic defini- tion, for instance Dirichlet's, requires function to be some determination of a variable in reference to a rule. As in any case either negatively or affirmatively determined in reference to a rule, the defined function cannot be the singular first predicate, andean only be the general any successive pred- icate, in which different successive predicates are connected. POSTULATE OF SCIENCE { ABSTRACT } ANALYSIS The solution of one case of Science { abstract } Analysis is first given or found in the shape of the single series o minimum maximum oo POSTULATE OF SCIENCE { FUNCTION } ANALYSIS The solution of one case of Science { function } Analysis is first given or found in the shape of the two connected series oo max. min. o min. max. c» NOTE One and the same degree in different places, for instance zero, has a certain singular character as proper to the single series, and a certain general character as common to the two connected series. In reference to the rule either plus or minus ± , every degree proper to the single series is indeter- minate, that is neither negatively nor af&rmatively deter- mined. In reference to the same rule any degree common to the two connected series is determinate, that is either negatively or affirmatively determined; in particular zero there as neither plus nor minus is negatively determined, and any other degree there as either plus or minus is affir- matively determined. The necessary and sufficient sign of zero as neither plus nor minus is only a little clear space in the case of zero corresponding to the space filled by the actual sign of either plus or minus in the case of any other degree. For example, in the whole of the notation o ± CD the clear space in question is in truth a sign to be read neither plus nor minus. Comparison and distinction of degrees in Science { abstract } Analysis belong in Science { abstract } Analysis. Comparison and distinction of degrees and signs in Science { function } Analysis belong in Science { function } Analysis. Resolution of all cases of Science { abstract } Analysis achieved in and through the integration of a differential equation de- rived from the postulated solution of the first case. minimum maximum o CX) min. max. o 00 min. max. intuition concept idea ideal of Science | abstract | Analysis synthesis 1 analysis I ^^^^^^ dialectic ui ^ minimum maximum o abstract ] mathematical 00 mm. max. l imi t minimum 1 ^i i^^ity maximum J * -' metaphysical { ^^^Zi::Z! } ^— ^^^^ logical! fj^ly"^^ I universality o gs Q ^ '-•' CO 10 Resolution of all eases of Science | function} Analysis acMeved in and through the integration of a differential equation de- rived from the postulated solution of the first case. 00 max. mm. mm. max. o =1= 00 ^jmin. [max. o ± 00 ^jmin. (.max. intiiitioii concept idea synthesis analysis dialectic ideal of Science | function | Analysis method P 00 nun. min. max. o CO m van- ^ able (s tMng miag. real no some o min. max. 00 individual general universal none one other one or other o function ^ g m I general any J immed. iindis. I thought [ concept distinct \ judgment \ mediate [ syllogism * reference of cognition to an object in general 11 NOTE Helpful to an understanding of the thing done on pages 10, 11, is a new use of the circle comparable to the classic use of the same by Euler in logic. Witness 80° O Ql o ^^ CD ^. § § p CD r1- O c-K CD o pi a^ P s Pj CD P p CD g P^ CD ^ ^ t3" a CD i CD ^ S CD cc CD ^ 13 s •r$ CD i— ' i- PJ CD O C5 ■ CD ^ O 05 CD CD CD ^ pj t^ Pj fcis CO o o B 2. 8 o 8 o cr5 cc 2 ^ H 00 ® O) '^'^ ^ ^ (S M s * ® • o 2 g S • ^ U 12 Also witness 80O O H- H- IQI O ^* *^ 2 <^ r> P CO CO 'P P p ^ P P OQ I Pj Pj p-o ® p PU 3 ^ Vj. p. CD P. P5 Th6 ¥el the who complet greesin ing the Analysis . universal J of developing the analytical expression Science | function | Analysis into a complete series, first becomes possible. The theory of functions in classic mathematics depends upon the two connected series of numbers oo . . . 3 (2 ^1 (0, 1, 2, 3 ... 00 where zero is neither plus nor minus, and any other num- ber is either plus or minus. Implicit in the two connected series of signed numbers is the variety { } 1 1 2, 3 ... [ =t CO J where the clear space, corresponding to the space filled by the actual sign of either plus or minus, is, in that corre- spondence, the necessary and sufficient sign of neither plus nor minus. The first comparison and distinction of the 14 numbers and signs conspicuous in the variety is recorded in the same notation as before in the case of t]ie corresponding variety of signed degrees on page 11. Therefore the notation expresses a common theory of function in respect of which classic mathematics and the present treatise are identical, however different they are in other respects. The notation in question variable imag. I } j individual real \ general universal function exhibits, in necessary and universal reference to function as any successive predicate, the sequence of one successive predicate conveniently read blank, and another suc- cessive predicate x, where the sequence of and x repre- sents any moment of the well-ordered logical determination of imaginary and real in necessary and universal reference to variable, apart from which, by definition, function is impos- sible. Since there is only one moment in the whole of the logical determination of imaginary, any moment of that determination is properly represented by the notation of that one in the place corresponding to the place of x repre- senting any moment of the logical determination of real. 15 POSTULATES OF SCIENCE { } ANALYSIS science! X } ANALYSIS I The solution of one case of Science { } Analysis is first given or found in the shape of a plane equilateral triangle the internal determination of which is only imaginary II The solution of one case of Science { individual } Analysis is first given or found in the shape of a plane equilateral triangle the internal determination of which is real in respect of only the middle point of the altitude m Resolution of all cases of Science {indi\T.dual} Analysis = solution of the first case of Science { general } Analysis. IV Resolution of aD. cases of Science {general} Analysis: solution of the first case of Science {universal} Analysis. HENCE by corresponding stages: 16 I Eesolution of all cases of Science { } Analysis achieved in and through the integration of a differential equation derived from the postulated solution of the first case. of * 1 / intuition / concept ideal /\idea of Science } Analysis / synthesis / analysis method / \ dialectic intuition A synthesis concept analysis ideal , Science idea /\ / / A dialectic , method of Analysis where the only imaginary predicate, as a part of speech, is an adjective which is read blank not only in two places picked out by small braces, but also in two corresponding places not so picked outj and where in 3 the clear space within the notation, concept analysis, and again at the right of the interior right hand brace, is a plane not first cognized in intuition as a plane pure and simple, but successively recognized in concept and in idea as the ground of the complete determination of the triangle as a whole and of the triangle as reproduced by degrees. 17 II Resolution, of aH cases of Science { individual } Analysis achieved in and through the integration of a differential equation derived from the postulated solution of the first case. ideal of individual Science intuition concept idea synthesis analysis dialectic intuition ideal of Science { individual } Analysis method synthesis concept ^ analysis A B AB X idea CD dialectic I! ABC-D method of individual Analysis 18 Ill Resolution of all cases of Science { general } Analysis achieved in and throngii the integration of a differential equation derived from the postulated solution of the first case, and sufBlciently distinguished as complex from prior differential equations as simple. C D A B AB X CD II ABC-D intuition concept idea synthesis analysis dialectic ideal of Science { general \ Analysis method synthesis method of general Analysis analysis dialectic ffi intuition eg concept Z D X Y multiplicand idea CD )4 altitude multiplier C D A B AB X base X II II ABC-D area product zZ D xX yY base = area -r- Yz alt. X XX i altitude = area -^ base 11 II II area = area^-=- area where the complex form of the differential equation in 3 is rendered necessary by the specialty of the particular Science \ X \ Analysis in question. Only through the cross refer- ence of the two equivalent formulas obtained from 2 to the same whole of possible Science { general } Analysis, could it be perfectly differentiated a priori. 19 IV a Derivation from the postulated solution of the first case, of a compound differential equation demanding the resolution of all cases, of Science { universal } Analysis. A c D A B AB D X y Z D X Y zZ D xX yY base multiplicand XX X XXX base = area 4- }^ altitude CD }4 altitude multiplier }i altitude = area 4- base II II II II II II ABC'D area product area = area^4- area single definite individual 2 manifold definable general synthetical analytical restrictive intuition definitive concept > ideal universal idea I of Science | universal | Analysis dialectical synthesis " analysis [ method " dialectic 3 Here belong the empty tables ABC which follow. They perfectly differentiate the whole of possible Science j universal } Analysis, and constitute a compound differen- tial equation demanding the resolution of all cases of that problem. 20 Pel o 'A < >; fe o o > S o s < syn. an. dia. synth. syn. an. dia. anal. syn. an. dia. dial. Me PHOD OF Universal Anali 21 'SIS s o i g > - syn. an. dia. synth.. syn. an. dia. 'anal. syn. an. dia. M ETHOD OF Universal Anai 22 .YSIS o » Q syn. an. dia. synth. syn. an. dia. anal. syn. an. dia. dial. Mi :thod of Universal Anal 23 YSIS IV b Essay to refer recognized moments of the demanded resolution of all cases of Science { universal } Analysis each to its proper place in the formula demanding the resolution. On the supposition that pure reason, constant as the faculty of Science = Analysis in all rationals of all times, but varied through all moments of the more and more definite and particular use of that faculty according to the rule function in different rationals of different times, has somewhere in some context already cognized every step in the solution of every case of Science { universal } Analysis, but has not yet recognized any step in the solution of any case in its proper place in the resolution of all cases; I propose to search out all and only the cognitions that are the content of that resolution, and arrange them each in that proper place as fixed for it a priori by the empty tables ABC. To be sure the task is not for only one rational, but for every one interested in the development as much as possible in himself of the same faculty that aforetime made the cognitions, and is now in his person caUed upon to recognize what it has itself in other persons already cognized according to a fixed and ascertained formula. As my own discovery of the required cognitions and reference of them to this or that place in the formula is sure and complete as regards the solution of at least the first or singular case of the general problem in question, so all that is wanting to the perfection of the demanded resolution of all cases will undoubtedly be found, if able men, and such as are ac- quainted with what is classic in the use of pure reason, will endeavor to recognize the missing cognitions by the general bat sufficient marks that relegate them to one or another place in the formula in correlation with one or another mo- ment of the singular solution. 24 a o 1 % 1 < <4^ < n >, >i '^ oQ NQ NO Nq AB X CD = ABCD base X Yz altitude = area multiplicand x multiplier = pr. base = area -=- % altitude yz altitude = area -=- base 11 II II area = area2-j- area < EH syn. an. dia. synth. syn. an. dia. anal. syn. an. dia. dial. Method of Univeusal An 25 ALYSIS . — ' — , --H « III ill eS Art 0,^3^ eS p^Ss 1-c •S -3 -B P.fr §2S I 111 §.1§ CO CO O "*;; cc "B pi cs » t^ 2 2 S-E u_ ill .2 Mcrt 5(-l ^S is a-rd rt> Pi rt .2-2 g 111 .2^ tPl II 111 1 II III O (D ® ^ rt p! =s':3 (0 eS'C fl as « Pl cS I Si m S ffi « rd P-Pl ^ -s •-a ■e S^ 'So S 2 rt ® ® cS S sjTi. an. dia. synth. syn. an. dia. anal. S eS ® ® ® pi's O ,2 o'g o ::3 ®«H A « ® li 2 =* p CO hS C V syn. an. . dia. dial, Method of Universal Analysis 26 NOTE Table A in its present state is filled to only a certain extent, bnt so secures separate consideration of necessary and suf- ficient evidence that the distinctions comprehended under the title method of universal Analysis are not arbitrary, but represent corresponding differences in reality and truth. The same distinctions in the same order just as truly repre- sent corresponding differences in the groups that fill table B ; but the correspondence, unmistakable in table A, is dif- ficult, but not too difiicult, to detect in table B. At the end of table B, in three conspicuous compartments, is laid down what was hitherto lacking, a mathematically well ordered curriculum of the sciences deduced from a mathematical definition of science. The unit of the curric- ulum is one group. In reference to the first group as pro- totypal, the eleven different successive groups are ectypal. The first group is classic mathematics, meaning arithmetic algebra geometry once as separate sciences, and again as combined and raised to a higher power in the calculus. The organ on of the extension of our knowledge in respect of the first ectypal group is the present treatise. It deserves to be caUed posterior mathematics in reference to classic mathe- matics as prior. The next two groups it is convenient and exact to caU dynamics, in analogy with Kant's separation of the categories connected with relation and modahty as dy- namical, from the categories connected with quantity and quality as mathematical. It is also convenient and in con- formity with classic usage to call the first four groups phys- ics, the next four logic, the last four ethics. In prior mathematics there is no generally accepted notion of algebra corresponding to the notion of arithmetic as sci- ence of number. Nevertheless function is here required to be the subject-matter of algebra by the nature and position of the indicia discursive magnitude and plurality referred to quantity, corresponding to the derivation of function whether from signed number or from signed degree. 27 Aritlimetical number is usually termed positive, or posi- tive with exception of zero, which makes it at least generally congruent with a part of algebraical number. But accord- ing to the indicia discrete magnitude and unity referred to quantity, number in arithmetic cannot be congruent at all with number in algebra, and can only be, in contradistinc- tion to algebraical number, unsigned, absolute, abstract, as at least one mathematician, J. W. A. Young, already teaches. Corresponding to number unsigned in arithmetic and signed in algebra is space unsigned in synthetical geometry and signed in analytical geometry. In logic the sjrmbol of equivalence expresses a thought which as a concept is such that any different concept is subordinated to it according to the logical series predicate predicate{-*-y of which the logical equation predicate = predicate! ?-?y^^^ is only a transformation. The same thought expressed in the series and again in the equation is yet again expressed in the remark that the concept equals is the identical ground of the complete determination of any different concept. In logic also the state of logical extent, in case every degree thereof is neither negatively nor affirmatively determined in reference to the rule ±, is sufficiently indicated by the notation, log. ex, unsigned. The state of the same, in case any degree thereof is either negatively or affirmatively de- termined in reference to the same rule, is sufficiently indi- cated by the notation {log. ex. }± where the clear space to the left of the left-hand brace, corresponding to the space fiUed by the actual sign of either plus or minus, is, in that correspondence, the necessary and sufficient sign of neither plus nor minus. In ethics, according to the demonstration on pages 3, 4, of what constitutes perfection in any cognition referred to 28 any object, the production by degrees of the symbol of equivalence beginning with the inspection of the cognition termed a plane in geometry, first gives reason in the person of the producer something to do that it can do in respect of every one of its faculties in their proper order and connection. Accordingly such a production of this symbol is the neces- sary and sufficient means to the end of first educating the whole of reason in that person to a certain extent. First education is here definitive of any further education; for of course, in the order of deduction, only through first and according to first does any further ever become possible. 29 Deacidified using the Bookkeeper process. Neutralizing agent: Magnesium Oxide Treatment Date: Sept. 2004 PreservatlonTechnologies