Gottfried Wilhelm Leibniz (July 1, 1646-November 14, 1716) was a German mathematician and philosopher.
Leibniz made contributions in a vast array of subjects, including: politics, law, ethics, theology, history, philosophy, philology and poetry.
Leibniz made major contributions to physics and technology, and is credited, along with Sir Isaac Newton, with the inventing of infinitesimal calculus.
According to Leibniz's notebooks, a breakthrough occurred on November 11, 1675, when he employed integral calculus for the first time to find the area under the graph of a function y=?(x).
Leibniz introduced several notations which continue to be used, such as the integral sign of an elongated S, from the Latin word summa, and the d used for differentials, from the Latin word differentia.
Leibniz's product rule of differential calculus is still called "Leibniz's law". The theorem that tells how and when to differentiate under the integral sign is called "the Leibniz integral rule."
Leibniz invented the binary system, the foundation of virtually all modern computer architectures.
In philosophy, he is remembered for optimism, which is his conclusion that the universe is, in a restricted sense, the best possible one God could have made. He was one of the three greatest 17th-century rationalists, along with René Descartes and Baruch Spinoza.
Gottfried Wilhelm Leibniz wrote in "On the Ultimate Origination of the Universe" (1697):
<You may well suppose the world to be eternal; yet what you thus posit is nothing but the succession of its states, and you will not find the sufficient reason in any one of them, nor will you get any nearer to accounting rationally for the world by taking any number of them together: the reason must therefore be sought elsewhere. Things eternal may have no cause of existence, yet a reason for their existence must be conceived.
Such a reason is, for immutable things, their very necessity or essence; while in the series of changing things, even though this series itself may be supposed a priori to be eternal, this reason would consist in the very prevailing of inclinations.
For in this case reasons do not necessitate (that is, operate with absolute or metaphysical necessity, so that the contrary would imply contradiction), but only incline.
Hence it is evident that even by supposing the world to be eternal, the recourse to an ultimate cause of the universe beyond this world, that is, to God, cannot be avoided.> 1646GL001
Gottfried Wilhelm Leibniz wrote in "Principles of Nature and Grace, Based on Reason" (1714):
<As for the rational soul, or mind...it is not only a mirror of the universe of created things, but also an image of the divinity. The mind not only has a perception of God's works, but it is even capable of producing something that resembles them, although on a small scale.
For to say nothing of the wonders of dreams, in which we effortlessly (but also involuntarily) invent things which we would have to ponder long to come upon when awake, our soul is also like an architect in its voluntary actions; and in discovering the sciences according to which God has regulated things (by weight, measure, number, etc...), it imitates in its realm and in the small world in which it is allowed to work, what God does in the large world.> 1646GL002
Gottfried Wilhelm Leibniz wrote in "The Principles of Nature and Grace, Based on Reason" (1714):
<Now this sufficient reason for the existence of the universe cannot be found in the series of contingent things....Although the present motion...arises from preceding motion, and that in turn from motion which preceded it, we do not get further however far we may go, for the same question always remains.
The sufficient reason, therefore, which needs not further reason, must be outside of this series of contingent things and is found in a substance which...is a necessary being bearing the reason for its existence within itself; otherwise we should not yet have a sufficient reason with which to stop.
This final reason for things is called God.> 1646GL003
--
American Quotations by William J. Federer, 2024, All Rights Reserved, Permission granted to use with acknowledgement.
Endnotes:
1646GL001.