On Infinite (1650-1674)

"I will prove that there are infinite worlds in an infinite world. Imagine the universe as a great animal, and the stars as worlds like other animals inside it. These stars serve in turn as worlds for other organisms, such as ourselves, horses and elephants. We in our turn are worlds for even smaller organisms such as cankers, lice, worms and mites. And they are earths for other, imperceptible beings. Just as we appear to be a huge world to these little organisms, perhaps our flesh, blood and bodily fluids are nothing more than a connected tissue of little animals that move and cause us to move. Even as they let themselves be led blindly by our will, which serves them as a vehicle, they animate us and combine to produce this action we call life." (Cyrano de Bergerac,"The Other World", 1657)

"Whatever we imagine is finite. Therefore, there is no idea or conception of anything we call finite. No man can have in his mind an image of infinite magnitude; nor conceive infinite swiftness, infinite time, or infinite force, or inmate power." (Thomas Hobbes, "Of Man", 1658)

"Man is equally incapable of seeing the nothingness from which he emerges and the infinity in which he is engulfed." (Blaise Pascal, "Pensées", 1670)

"What is man in nature? A Nothing in comparison with the Infinite, an All in comparison with the Nothing, a mean between nothing and everything. Since he is infinitely removed from comprehending the extremes, the end of things and their beginning are hopelessly hidden from him in an impenetrable secret; he is equally incapable of seeing the Nothing from which he was made, and the Infinite in which he is swallowed up." (Blaise Pascal, "Pensées", 1670) 

"Nature is an infinite sphere of which the center is everywhere and the circumference nowhere." (Blaise Pascal, "Pensées", 1670)

"We know that there is an infinite, and we know not its nature. As we know it to be false that numbers are finite, it is therefore true that there is a numerical infinity. But we know not of what kind; it is untrue that it is even, untrue that it is odd; for the addition of a unit does not change its nature; yet it is a number, and every number is odd or even (this certainly holds of every finite number). Thus, we may quite well know that there is a God without knowing what He is." (Blaise Pascal, "Pensées", 1670)

"And just as the advantage of decimals consists in this, that when all fractions and roots have been reduced to them they take on in a certain measure the nature of integers, so it is the advantage of infinite variable-sequences that classes of more complicated terms (such as fractions whose denominators are complex quantities, the roots of complex quantities and the roots of affected equations) may be reduced to the class of simple ones: that is, to infinite series of fractions having simple numerators and denominators and without the all but insuperable encumbrances which beset the others." (Isaac Newton, "De methodis serierum et fluxionum" ["The Method of Fluxions and Infinite Series"], 1671)

"From this it follows that two infinites can be equal, just as in an infinite series of pairs there necessarily exist two series of unities completely identical, provided that a unity corresponds always to another unity, so that there is no excess or defect in one or the other series, which corresponds, in turn, to the definition of equality. From here you also grasp that one can properly speak of 'quantity' in the sense we have explained but not in that ordinary way in which by 'quantity' is usually understood a certain determinate number in its kind corresponding to another in the number of units." (Emmanuel Maignan, 1673)

"The same thing is confirmed through the previous proposition since, for example, an infinite collection from which units can be subtracted (not only ten but infinitely many) while the collection remains infinite is, obviously, infinitely greater before the subtraction takes place than after; thus, since the collection does not cease being infinite after the subtraction, the infinite will be, as such, infinitely smaller than it was earlier. You could say that this is in conflict with the generally accepted thesis that holds that the terms 'greater' and 'smaller' can only apply to finite quantities but not to infinite quantities; or, at least, that they can be applied to infinite quantities only in a very improper way. I reply that this idea has the following feature, namely that it is widespread. This fact notwithstanding, I would say, with permission, that it also has this other feature, namely that its ground is nothing else but a false notion of infinity. Moreover, the advantage it offers, which consists in apparently solving some difficulties that are usually put forth by denying that 'greater' and 'smaller' are properties that can be predicated of infinity, does not subsist for it ends up not resolving the difficulties." (Emmanuel Maignan, 1673)

"In practical life we are compelled to follow what is most probable; in speculative thought we are compelled to follow truth. […] we must take care not to admit as true anything, which is only probable. For when one falsity has been let in, infinite others follow." (Baruch Spinoza, [letter to Hugo Boxel], 1674)