The demonstration in this proposition is exceptionally difficult and unusual as such. First, the demonstrations typically employ what has already been proven. But the first reference to a prior proposition is related to attributes, yet introduces a new concept - that thought is an attribute. The specific identity of the attributes has not yet been established in Spinoza's typically rigorous build. Next, the Latin is vague because both the Latin words for thought (cognatio) and for idea (idea) are feminine. As Spinoza builds his first demonstration, translators experience enormous confusion and, in my opinion, are wrong here. The problems start with the sentence Verum quatenus ipsa ideam Dei habet, finita supponitur. In this sentence, translating the first half as "Indeed, insofar as it has the idea of God," "it" (ipsa) is clearly referring to thought rather than idea since the latter would be redundant. All translators agree.
The second half of the sentence is where interpretations of "it" diverge in the attempt to translate what reads as "it is placed as finite." Almost all continue to make "it" refer to thought. However, this makes no sense at all. He has just clearly stated in the prior sentence that thought is an attribute and attributes are infinite. If this had been his rendering, Spinoza would have typically concluded "and this is absurd." But he didn't. He moves into an elaborate discussion to address the exceptional difficulty of a piece of an attribute - like idea within thought - that is not finite. So my translation is correct and focuses on a mode or a piece of an attribute which is not finite. This sets up an issue - all modes are not alike as those which follow from the absolute nature of an attribute (which is what this proposition is about) differ from those which do not follow from the absolute nature of an attribute.
The rest of the first proof of his demonstration moves into complete unreadability by following the mistranslation. Most commentators throw up their hands and just skip the nuances of the demonstration. That is a shame because Spinoza explores a critical issue here of movement from the infinite to the finite. The next sentence employs the rarely utilized D2 with its specificity of finiteness and it becomes critical - bodies limit bodies and thoughts limit thoughts - as specific finite things within infinite attributes. So here we have the issue that a finite idea must be limited by another finite idea as a finite thought (as opposed to the attribute of thought). However, a close reading highlights this special category of modes that almost function with attribute-like characteristics, as they inhere in the nature of each mode within the attribute. Conversely, they can be conceived of as a special category within attributes that have modal characteristics. As a result, this liminal level appears to play an important role in moving from infinite to finite.Omnia quæ ex absoluta natura alicujus attributi Dei sequuntur, semper et infinita existere debuerunt sive per idem attributum æterna et infinita sunt.
Translated as,
All things which follow from the absolute nature of any of the attributes of God must have always existed as infinite or through the same attribute are eternal and infinite.
DEMONSTRATIO: Concipe si fieri potest (siquidem neges) aliquid in aliquo Dei attributo ex ipsius absoluta natura sequi quod finitum sit et determinatam habeat existentiam sive durationem exempli gratia ideam Dei in cogitatione. At cogitatio quandoquidem Dei attributum supponitur, est necessario (per propositionem 11) sua natura infinita. Verum quatenus ipsa ideam Dei habet, finita supponitur. At (per definitionem 2) finita concipi non potest nisi per ipsam cogitationem determinetur. Sed non per ipsam cogitationem quatenus ideam Dei constituit; eatenus enim finita supponitur esse : ergo per cogitationem quatenus ideam Dei non constituit, quæ tamen (per propositionem 11) necessario existere debet. Datur igitur cogitatio non constituens ideam Dei ac propterea ex ejus natura quatenus est absoluta cogitatio, non sequitur necessario idea Dei (concipitur enim ideam Dei constituens et non constituens). Quod est contra hypothesin. Quare si idea Dei in cogitatione aut aliquid (perinde est quicquid sumatur quandoquidem demonstratio universalis est) in aliquo Dei attributo ex necessitate absolutæ naturæ ipsius attributi sequatur, id debet necessario esse infinitum; quod erat primum. Deinde id quod ex necessitate naturæ alicujus attributi ita sequitur, non potest determinatam habere existentiam sive durationem. Nam si neges, supponatur res quæ ex necessitate naturæ alicujus attributi sequitur, dari in aliquo Dei attributo exempli gratia idea Dei in cogitatione eaque supponatur aliquando non exstitisse vel non exstitura. Cum autem cogitatio Dei attributum supponatur, debet et necessario et immutabilis existere (per propositionem 11 et corollarium II propositionis 20). Quare ultra limites durationis ideæ Dei (supponitur enim aliquando non exstitisse aut non exstitura) cogitatio sine idea Dei existere debebit; atqui hoc est contra hypothesin; supponitur enim ex data cogitatione necessario sequi ideam Dei. Ergo idea Dei in cogitatione aut aliquid quod necessario ex absoluta natura alicujus attributi Dei sequitur, non potest determinatam habere durationem sed per idem attributum æternum est, quod erat secundum. Nota hoc idem esse affirmandum de quacunque re quæ in aliquo Dei attributo ex Dei absoluta natura necessario sequitur.
Translated as,
Conceive, if it is possible, (if indeed you deny [it]) something in some attribute of God that follows from absolute nature itself which may be finite and may have a determinate existence or duration, for example, the idea of God in thought. But thought, since it is placed as an attribute of God, is necessarily (by P11) infinite by its nature. Indeed, insofar as [infinite thought as attribute] itself has the idea of God, it [the idea of God] is placed as finite. But (by D2) a finite [thought] is not able to be conceived unless it [a finite thought] is limited by [a finite] thought itself. But not by [infinite] thought [as attribute] itself insofar as it [infinite thought as attribute] constitutes the idea of God, but so far as it [a thought] is placed to be finite. Therefore, by [a finite] thought insofar as [infinite thought as attribute] does not constitute the idea of God, which nevertheless (by P11) [the idea of God] must exist necessarily. Consequently, there is [infinite] thought [as attribute] not constituting the idea of God and for that reason, from its [thought's] nature insofar as it is absolute thought, the idea of God does not follow necessarily (for it [infinite thought as attribute] is conceived as constituting and not constituting the idea of God). This is contrary to hypothesis. So if the idea of God in thought or anything else (in the same manner is anything assumed since the demonstration is universal) following in some attribute of God from the absolute nature of the attribute itself, it is necessarily infinite. This was the first thing.
Next that which follows from the necessity of the nature of some attribute, can not have a determined existence or duration. For if you would deny this, let a thing be placed which follows from the necessity of the nature of some attribute, that is in some attribute of God, for example the idea of God in thought and let it be placed in some way that it did not exist or would not exist. Moreover, since thought is placed as an attribute of God, it must both necessarily and immutably exist (by P11 and P20C2). For this reason, final limits of the duration of the idea of God (for it is placed that for some reason it did not exist and would not exist), thought would exist without the idea of God. But this is contrary to hypothesis. For let it be placed that from thought given follows necessarily the idea of God. Therefore, the idea of God in thought or something which follows necessarily from the absolute nature of some attribute of God must follow that it cannot have a determined duration but is eternal through the same attribute. This noted follows that this same must be affirmed concerning whatsoever thing which is in some attribute of God which follows from the absolute nature of God.
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