Spinozas Ethics is organised into a deductive format that reveals the necessary connections between the propositions. The geometrical method contributes to a necessary and indubitable outcome that God necessarily exists and this will be an absolute truth. The Ethics geometrical method is based after Euclids work where the demonstration of every proposition uses definitions, axioms and only propositions that have been already established. I will be focusing on whether the definitions are real or nominal as this is the main source of controversy on what the geometrical method contributes to the existence of God. I will be arguing that the definitions are real and constructive therefore the geometrical method contributes to a logical argument for the existence of God.
Proposition 11 is the first proposition where God is mentioned and this proposition contains three demonstrations for God's existence. The first demonstration is the ontological argument which tries to deduce the existence of God from concepts. It states that it is impossible to conceive of God not existing. This is because God is defined as 'a substance consisting of an infinity of attributes' (ID6) and it has already been proven that 'it pertains to the nature of a substance to exist' (IP7). Therefore because of the geometrical method all Spinoza has to do is substitute 'God' for 'substance' to show that God necessarily exists. To say that God does not exist is to say that it is not in God's essence to exist but this is absurd as we have seen 'the nature of a substance is to exist' and God is a substance.
Spinoza has already demonstrated that 'if things have nothing in common with one and other, one of them cannot be the cause of the other' (IP3), 'in nature there cannot be two of the same substances of the same nature or attribute' (IP5) and 'two substances with different attributes have nothing in common' (IP2). Thus it logically follows that 'one substance cannot be produced by another substance' (IP6). As 'a substance is prior to its affections', this means that they cannot produce a substance. Therefore 'it pertains to the nature of a substance to exist' (IP7). By definition God is a substance which means it pertains to the nature of God to exist so God necessarily exists. As it is evident from this example, the geometrical method allows Spinoza to state God's existence as an unquestionable matter of fact without room for reasonable disagreement with his logical conclusion. It contributes the ability to show persuasively how one proposition follows from another, leading to clear and distinct necessary truths which are not contingent but are absolute and objective. It shows a 'rigid mathematical necessity' which doesn't rely on our experiences and eliminates human ignorance. Moreover nothing relies on unproven claims or assumptions that might make the argument for the existence of God incorrect.
The geometrical method for the existence of God is undeniable only if we agree with the definitions and axioms. If we do then we will find it difficult (but not impossible) to not agree with Spinoza's conclusion. I will focus on Spinoza's definitions as they are the 'bedrock' of his method. Whether the geometrical method contributes to a logical argument for the existence of God depends on whether Spinoza's definitions are nominal or real. Nominal definitions are what is meant by a word or thought in a concept thus they can be nothing about reality. They are merely stipulative, have no truth value and can be arbitrary. Whereas real definitions 'explain a thing as it is outside of the understanding' , they are actually defining something. They have truth values as they are accurate representations of a thing. If the definitions are nominal then the geometrical method for God's existence is weak as the ontological argument relies on the definition of God yet this definition will not be saying anything about God in reality. But I believe that Spinoza's definitions are real definition and define something that actually is. Therefore the geometrical method contributes to a logical argument for the existence of God.
One claim by R. J. Delahunty is that the definitions must be nominal not real if they are to be intelligible. Delahunty argues that the definitions have to be nominal because of the form of the definitions, for example 'By ... I understand...' Another claim is mentioned in Spinoza's letter to Simon De Vries, which states that the truth of the conclusion that one deduces from the definitions is not changed by the definitions being arbitrary or even false. This could mean that Spinoza is trying to deduce God's existence from some basic but not certain truths. I will argue against these points as I believe we should not look at the geometrical method of the Ethics in this way as it does contribute to God's existence.
To refute the first claim about the definitions being nominal because of their phrasing I will turn to P. Basile who disputes Delahunty's argument. He believes that Spinoza has done this so that we think understand the definitions in the same way as he does. Basile refers to Spinoza's second letter to Oldenburg where Spinoza explains that is possible to use real definitions and still phrase them in the way he does:
'The definition of God is true appears from the fact, that by God we mean a Being supremely perfect and absolutely infinite.'
Thus the definition of God as having infinite attributes is not in oppositions to our definition of God. Basile concludes that the second letter suggests 'Spinoza's definitions would be an analysis of the ordinary conception of God as an infinite being.' Therefore Spinoza's definitions are real and give a clarified version of our idea of God, contributing to a strong argument for the existence of God.
To refute the second claim I will refer to Steven Nadler. He believes that this is not the correct interpretation of the Ethics as Spinoza is demonstrating truth; 'the Ethics is about reality.' The definitions are not nominal as the arguments are not that if you 'assume for the sake of the argument' what the definitions mean then God necessarily exists. Rather he is saying that this is how the definitions 'should be understood, if defined truly' therefore God necessarily exists.
Spinoza in Treatise on the Emendation of the Intellect states that mathematicians know the real definition of a figure when they are able to construct a figure; this is a constructive definition. Spinoza describes the notion of a sphere to a student by saying that 'a semicircle is rotated around a centre.' While a sphere has not actually been constructed in this way, imagining it has can help us understand the mathematical definition of a sphere. If this is applied to the definition of God then it could be seen as constructive because it demonstrates which concepts we need to construct together to achieve 'a correct metaphysical notion of the Diety, one that is wholly purged of misleading association inherited from tradition, prejudices, or false metaphysics.' This idea would account for phrases such as 'by...I understand...' This would also be compatible with the idea that to understand God is to have clear concepts of a real thing. Thus a constructive definition leads us to understand what something is. Just as the semicircle helps us to understand the sphere, the definition helps us to understand God. As the geometrical method uses these constructive definitions, through proposition 11 and the definitions of substance and God, the reader is helped to understand that God necessarily exists.
Moreover, Spinoza shows the difference between a nominal definition of a circle 'a figure in which the lines drawn from the centre to the circumference are equal' and constructive as 'the figure that is described by any line of which one end is fixed and the other movable.' The constructive definition not the nominal definition shows us how a circle can be constructed and because of this we can derive at its essential properties. Spinoza believes that this can be applied to reality and that we can have a real idea about something through its causes:
'Knowledge of the effect is nothing but acquiring a more perfect knowledge of its causes.'
Henry Allison states that 'either we find ourselves involved in an infinite regress, which would lead to hopeless scepticism, or the chain of reasoning must be grounded in a single principle.' This principle is the causa-sui as everything must be explained by it and its 'essence must involve existence.' Therefore the geometrical method contributes to the definition of God and this definition includes God necessarily existence.
In conclusion, I believe that Nadler and Basile arguments for the definition of the Ethics to be real and constructive are correct. As they are real definition, the ontological argument in proposition 11 is based in reality therefore contributes to a logical argument for the existence of God. As the definitions are constructive, the geometrical method contributes to constructing the right concepts together to have a clear concept of a God who exists. Moreover, because the definitions are constructive the geometrical method contributes to us deriving at God's essential properties of God, one of these being existence.