Gottfried Leibniz was a contemporary of Spinoza (they are called the two greatest rationalists after Descartes – read correctly either way) who had reviewed the Ethics but did not agree philosophically on the grounds that it contradicted Orthodox belief.
His writing is pretty impenetrable – am probably taking some liberties with his views. As an aside this liberty is a noted triumph of postmodern scholarship – overdetermination, the attack on Essentialism…to be elaborated later…
In his work First Truths he vigorously defends the Aristotelian/Scholastic view that if something is true than its negation is false. This view was seen as the foundation of all logic and knowledge in the so-called classical view of philosophy.
He derives this from I believe a mathematical appreciation for the world. In First Truths he discusses the uniqueness of all living things. He says there are no two things in nature which differ only numerically:
So perfect similarity occurs only in incomplete and abstract concepts, where matters are conceived, not in their totality but according to a certain single viewpoint, as when we consider only figures and neglect the figured matter. So geometry is right in studying similar triangles, even though two perfectly similar material triangles are never found. And, although gold or some other metal, or salt, and many liquids, my be taken for homogeneous bodies, this can be admitted only as concerns the senses and not as if it were true in an exact sense.
So this would seem to contradict Spinoza, seemingly suggesting that a single substance does not exist. I suspect one of Spinoza’s most radical views is that man is not separable from God or nature except by human conception alone. This speaks to the power of human conception, which is remarkable – creation itself could be a name for God. Mimicking this divine creation is a great ability and I want to get back to this in later posts.
According to Leibniz: God has equipped both body and soul from the beginning with such great wisdom and workmanship that, through the original constitution and essence of each, everything which happens in one corresponds perfectly and automatically to whatever happens in the other. I call this the hypothesis of concomitance. This is true of all the substances in the whole universe but is not perceptible in all as it is in the soul and body.
I find his final sentence brilliant. Even back in 1680 – the estimated date of writing – Leibniz understood the fundamental properties of space and matter…the interconnectedness of all things. His faith in the “workmanship” of body and soul leads him to believe that what we perceive to be separate and is necessarily separate. At the same time he goes on to say that there is no vacuum (because that would mean different parts of empty space were perfectly similar and congruent with each other and could not by themselves be distinguished. So they would differ in number alone, which would be absurd) and no atoms (every body, no matter how small, is actually subdivided…every particle in the universe contains a world of infinitely many creatures – I think he is presciently challenging the notion of the atom as conceived in his era, the final smallest building block of matter), and even doesn’t leave time untouched (time may prove to not to be a thing, in the same way as space).
He asserts that each individual substance involves the whole universe in its perfect concept, and that which has existed or will exists must be included in a perfect or complete concept.
Although Leibniz disagreed with Spinoza, on what seems on the surface to be moral grounds, their philosophies are not contradictory at all, unless one believes in contradictions. Haha.
Leibniz seems to insist that all substance is different and varied. Thus, zero is just really really small and the infinite is really really big. I find an interesting correlation between this thought and the idea of postmodernism which supposes a theoretical or infinite amount of “voices” and the new criticism which will emphasize an actual very large amount of “voices.”
At the same time Spinoza says that properties depend on substance. Substance is independent, and to be independent it must be not limited – infinite – and thus there is a single substance God.
We are mere modifications of this substance. Here Leibniz seems to agree. Each created individual substance, indeed, are different expressions of the same universe and of the same universal cause, God. But those expressions vary in perfection as do different representations or perspectives of the same city seen from different points.
It would seem the difference lies in the interpretation of “varying in perfection.” Both agree that perspective is important. It appears that Spinoza gives less weight to the validity of human perception as a sort of zero value compared to God.
The real clash seems to come from their application of their ideas in the political realm. Spinoza was a champion of the secular state, supporting ideas such as free speech and democracy long before the French Revolution.
Leibniz on the other hand, quoting from this site: “So why do bad things happen? They must either not be bad, or else somehow they must be necessary. If a better world were possible, God would have made it. This must be the best of all possible worlds.
Time and space have only a relative reality. If time were real, when would it have begun? And why would God have started it then rather than some other time? There can be no reason, but nothing happens for no reason. The same goes with space. Why would God put space here rather than there? Space cannot be an absolutely existing thing. It can only be a relative thing, a matter of a certain kind of relation between physical objects.”
The notion of the best of all possible worlds was especially upsetting to Voltaire, who wrote an entire short novel, Candide, to chide and ridicule this point by documenting plain cases of suffering in the world, and then excusing it blithely and sardonically as God’s will and the best of all possible worlds – a notion that likely helped to boost theocracies and aristocracies lingering from the Medieval period.
It can boiled down to cause and effect – a useful oversimplification would be a comparison between: the ends justifying the means (Leibniz – God is all powerful and has created all motions and objects and set them in a pre-established harmony, thus whatever happens is God’s will so if a tyrant kills all his people God could not have planned it any other way…this returns to Aristotle and Leibniz makes his case by discerning between contingent and necessary truths…more later); the alternative is that the ends are the means (God is actually the clockwork, that which allows existence to exist and is infinite in its freedom and possibility), which is more Spinoza’s contention. Therefore to act in a good and just manner is to adequately reflect a conception of God, instead of succumbing to our hungers which is the mark of an incomplete understanding of our bodies and intellect and their relation to the universe. On the other hand to Leibniz nothing is the same except for the motions which drive individual objects. These motions are uniform (math): The cause must be inferable from the effect.
Of course our job is to see how these can both be true.
Active Philosophy 
Leibniz was actually a very talented and accomplished mathematician. In fact, he independently (from Isaac Newton) developed differential calculus, which is probably the most widely applied mathematics in physics. Furthermore the notation he developed is the notation we use today. If you read his wikipedia page you will find that his range of accomplishments are very impressive (philosophy to mathematics).
This post made me think of an important mathematical theory from two other important mathematicians: Euler-Maclaurin Formula.
Often times in mathematics (and scientists in general ), we want to compute a summation of discrete quantities, which is very similar (conceptually, but not necessarily computationally) to doing an integral. Integrals, in essence, are summations over continuous quantities.
A widely used method for computing summations is to approximate them as being continuous, or by approximating them with an integral. This is ‘justified’ when the difference between a given discrete value and the next in the summation is very small. The degree of accuracy more or less scales with how small this difference is.
The Euler-Maclaurin Formula is essentially a formula that computes the error of approximating the summation with an integral. If the approximations is applied correctly this ‘error’ is very small.
This is relevant to Leibniz discussion of similar triangles. They exist in math but not in nature. And although nature may mimic the math (or vice versa) there is always some level of approximation.
Wonder if Einstein read this before coming up with Relativity
“Time and space have only a relative reality. If time were real, when would it have begun? And why would God have started it then rather than some other time? There can be no reason, but nothing happens for no reason. The same goes with space. Why would God put space here rather than there? Space cannot be an absolutely existing thing. It can only be a relative thing, a matter of a certain kind of relation between physical objects.”
Did you write this or Leibniz?
What seems unclear about the last sentence is that physical object ‘take up’ or exist within space. Not sure… I’ll think about his some more.
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“Space cannot be an absolutely existing thing. It can only be a relative thing, a matter of a certain kind of relation between physical objects.”
It comes from a website that is probably doing his ideas injustice.
I think this sentence depends on what the popular definition of space for his time period is. But I’ll take a stab and think he is debating the uniformity of space as being impossible in reality just as an absolutely pure circle. This returns to his idea that individual things exist but group themselves together. As in you can cut here or there in a line, bisect or trisect it, and the divisions are real although the location of the incision may be arbitrary.