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| − | ==The intermediate value theorem==
| + | Each statement of a syllogism is one of 4 types, as follows: |
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| − | As noted above, Gauss' 1816 proof of the fundamental theorem of algebra relied on the intermediate value theorem. The following statement of the theorem, used to determine intervals in which a function has roots, is sometimes termed Bolzano's Theorem:
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| − | :if f(x) is a continuous function of a real variable x and if f(a)<0 and f(b)>0, then there is a c between a and b such that f(c)=0. | + | |- |
| − | Indeed, it was Bolzano's insight that the theorem, though very plausible, needed to be proved.
| + | ! Type !! Statement !! Alternative |
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| − | In his paper of 1817, Bolzano undertook to prove the theorem, which he stated in terms of the roots of a polynomial equation in one real variable as follows:<ref>Russ p. 181</ref>
| + | | style="text-align: center;" | '''A''' || '''All''' $A$ '''are''' $B$ || |
| − | :If a function of the form
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| − | ::$x^n + ax^{n-1} + bx^{n-2} + ... + px + q$
| + | | style="text-align: center;" | '''I''' || '''Some''' $A$ '''are''' $B$ || |
| − | :in which $n$ denotes a whole positive number, is positive for $x = \alpha$ and negative for $x = \beta$, then the equation | + | |- |
| − | ::$x^n + ax^{n-1} + bx^{n-2} + ... + px + q = 0$
| + | | style="text-align: center;" | '''E''' || '''No''' $A$ '''are''' $B$ || (= '''All''' $A$ '''are not''' $B$) |
| − | :has at least one real root lying between $\alpha$ and $\beta$. | + | |- |
| − | In his prefatory remarks, Bolzano discussed in detail previous proofs of the intermediate value theorem. Many of those proofs (alike with Gauss' 1799 proof of the fundamental theorem of algebra) depended "on a truth borrowed from geometry." Bolzano rejected all such proofs in totality and unequivocally:<ref>Russ p. 160</ref>
| + | | style="text-align: center;" | '''O''' || '''Not All''' $A$ '''are''' $B$ || (= '''Some''' $A$ '''are not''' $B$) |
| − | :It is an intolerable offense against correct method to derive truths of pure (or general) mathematics (i.e., arithmetic, algebra, analysis) from considerations which belong to a merely applied (or special) part, namely, geometry.... A strictly scientific proof, or the objective reason, of a truth which holds equally for all quantities, whether in space or not, cannot possibly lie in a truth which holds merely for quantities which are in space. | + | |} |
| − | Other proofs that Bolzano examined and rejected were based "on an incorrect concept of continuity":
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| − | :No less objectionable is the proof which some have constructed from the concept of the continuity of a function with the inclusion of the concepts of time and motion.... No one will deny that the concepts of time and motion are just as foreign to general mathematics as the concept of space.
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| − | Bolzano here arriving at the first mathematical achievement of his paper, stated a formal definition of the continuity of a function of one real variable, as follows:<ref>Russ p. 162</ref>
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| − | :If a function $f(x)$ varies according to the law of continuity for all values of $x$ inside or outside certain limits, then if $x$ is some such value, the difference $f(x + \omega) - f(x)$ can be made smaller than any given quantity provided $\omega$ can be taken as small as we please.
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| − | In the subsequent process of proving the main theorem stated above, Bolzano achieved the following additional mathematical results:<ref>Russ p. 157</ref>
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| − | * he introduced the criterion for the (pointwise) convergence of an infinite series, as follows:
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| − | : If a series of quantities
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| − | ::$F_1x$, $F_2x$, $F_3x$, . . . , $F_nx$, . . . , $F_{n+r}x$, . . . | |
| − | :has the property that the difference between its.$n$th term $F_nx$ and every later term $F_{n+r}x$, however far from the former, remains smaller than any given quantity if $n$ has been taken large enough, then there is always a certain constant quantity, and indeed only one, which the terms of this series approach, and to which they can come as close as desired if the series is continued far enough.
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| − | :As noted previously, Bolzano was here able to establish only the ''plausibility'' of the criterion. A proof of its sufficiency had to await the definition of the real number field.
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| − | * he stated and proved the [[Bolzano-Weierstrass theorem]] in its original form:
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| − | :If a property $M$ does not belong to all values of a variable $x$, but does belong to all values which are less than a certain $u$, then there is always a quantity $U$ which is the greatest of those of which it can be asserted that all smaller $x$ have property $M$
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| − | * he stated and proved the intermediate value theorem in a form that is now sometimes called Bolzano's theorem, which Bolzano himself believed to be "a more general truth" than the main theorem he set out to prove:
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| − | :If two functions of $x$, $f(x)$ and $g(x)$, vary according to the law of continuity either for all values $x$ or only for those which lie between $\alpha$ and $\beta$, and if $f(\alpha) < g(\alpha)$ and $f(\beta) > g(\beta)$, then there is always a certain value of $x$ between $\alpha$ and $\beta$ for which $f(x) = g(x)$
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| − | Finally, he stated and proved the main theorem noted at the outset above.
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| − | ==Notes==
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| − | <references/>
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| − | ==Primary sources==
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| − | * Bolzano, Bernard (1817). ''Rein analytischer Beweis des Lehrsatzes, dass zwischen je zwey Werthen, die ein entgegengesetzes Resultat gewahren, wenigstens eine reelle Wurzel der Gleichung liege''. ["Purely analytic proof of the theorem that between any two values which give results of opposite sign there lies at least one real root of the equation"]. Prague.
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| − | ==References==
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| − | * Russ, S. B. "A Translation of Bolzano's Paper on the Intermediate Value Theorem," ''Historia Mathematica'' 7 (1980), 156-185.
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