"Every word in language serves to designate an idea and some of them even complete propositions. Therefore, it is only natural to suppose that each idea is composed of at least as many parts as there are words in its expression." (Bernard Bolzano, "Wissenschaftslehre" ["Theory of Science"], 1837)
"One very important genus of complex ideas that we encounter everywhere are those in which the idea of collection (Inbegriff ) appears. There are many types of the latter [...] I must first determine with more precision the concept I associate with the word collection. I use this word in the same sense as it is used in the common usage and thus understand by a collection of certain things exactly the same as what one would express by the words: a combination (Verbindung) or association (Vereinigung) of these things, a gathering (Zusammensein) of the latter, a whole (Ganzes) in which they occur as parts (Teile). Hence the mere idea of a collection does not allow us to determine in which order and sequence the things that are put together appear or, indeed, whether there is or can be such an order. [...] A collection, it seems to me, is nothing other than something complex (das Zusammengesetztheit hat)." (Bernard Bolzano, "Wissenschaftslehre" ["Theory of Science"], 1837)
"The most plausible way to [conceive of the relation between grounding and causality] is that somehow those truths that state the existence of the properties of a cause be considered as the ground, and those that concern the existence and the properties of the effect be considered as the consequence." (Bernard Bolzano, "Wissenschaftslehre" ["Theory of Science"] ,1837)
"[a set is] an embodiment of the idea or concept which we conceive when we regard the arrangement of its parts as a matter of indifference." (Bernard Bolzano, 1847)
"Already within the domain of those things which do not have any pretension of reality, but only of possibility, there indisputably are sets that are infinite. The set of propositions and truths in themselves is infinite, as one can easily see." (Bernard Bolzano, "Paradoxien des Unedlichen" ["Paradoxes of the Infinite"], 1851)
"But rather they are able, in spite of that relationship between them that is the same for both of them, to have a relationship of inequality in their pluralities, so that one of them can be presented as a whole, of which the other is a part. An equality of these pluralities may only be concluded if some other reason is added, such as that both multitudes have exactly the same determining grounds, e.g. they have exactly the same way of being formed." (Bernard Bolzano, "Paradoxien des Unedlichen" ["Paradoxes of the Infinite"], 1851)
"Even in the realm of things which do not claim actuality, and do not even claim possibility, there exist beyond dispute sets which are infinite. The set of all ‘absolute propositions and truths' is easily seen to be infinite." (Bernard Bolzano, "Paradoxien des Unedlichen" ["Paradoxes of the Infinite"], 1851)
"Even with the examples of the infinite considered so far it could not escape our notice that not all infinite multitudes are to be regarded as equal to one another in respect of their plurality, but that some of them are greater (or smaller) than others, i.e. another multitude is contained as a part in one multitude (or on the contrary one multitude occurs in another as a mere part).This also is a claim which sounds to many paradoxical." (Bernard Bolzano, "Paradoxien des Unedlichen" ["Paradoxes of the Infinite"], 1851)
"If they [mathematicians] find a quantity greater than any finite number of the assumed units, they call it infinitely great; if they find one so small that its every finite multiple is smaller than the unit, they call it infinitely small; nor do they recognise any other kind of infinitude than these two, together with the quantities derived from them as being infinite to a higher order of greatness or smallness, and thus based after all on the same idea." (Bernard Bolzano, "Paradoxien des Unedlichen" ["Paradoxes of the Infinite"], 1851)
"[...] from that circumstance alone we are not allowed to conclude that both sets, if they are infinite, are equal to each other with respect to the multiplicity of their parts (that is, if we abstract from all differences between them); [...] Equality of those multiplicities can only be inferred when some other reason is added, for instance that both sets have absolutely equal grounds of determination, i.e., that their mode of formation is absolutely equal." (Bernard Bolzano, "Paradoxien des Unedlichen" ["Paradoxes of the Infinite"], 1851)
"Space and time - which again do not belong to the domain of the actual, though they can be determinations of the actual - form a very important category of infinitely great quantities." (Bernard Bolzano, "Paradoxien des Unedlichen" ["Paradoxes of the Infinite"], 1851)
"The existence of infinite sets, at least with non-actual members, is something which I now regard as sufficiently proved and defended; as also, that the set of all absolute truths is an infinite set." (Bernard Bolzano, "Paradoxien des Unedlichen" ["Paradoxes of the Infinite"], 1851)
"The very word in finite shows that we put the infinite into contrast with the merely finite. Again, the derivation of the former name from the latter betrays the additional fact that we consider the idea of the infinite to arise from the idea of the finite by, and only by, the adjunction of a new element; for such in fact is the abstract idea of negation." (Bernard Bolzano, "Paradoxien des Unedlichen" ["Paradoxes of the Infinite"], 1851)
"Therefore both multitudes have one and the same plurality, as one can also say, equal plurality. Obviously this conclusion becomes void as soon as the multitude of things in A is an infinite multitude, for now not only do we never reach, by counting, the last thing in A, but rather, by virtue of the definition of an infinite multitude, in itself there is no last thing in A, i.e. however many have already been designated, there are always others to designate." (Bernard Bolzano, "Paradoxien des Unedlichen" ["Paradoxes of the Infinite"], 1851)
"By a 'Satz an sich' [senternce in itself] I mean any statement whatever to the effect that something is or is not, irrespective of whether the statement be true or false, irrespective of whether any person ever formulated it in words, and even irrespective of whether it ever entered into any mind as a thought." (Bernard Bolzano)
"My special pleasure in mathematics rested particularly on its purely speculative part." (Bernard Bolzano)
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