I've tried many times to understand Spinoza's philosophy, but each time I find myself unable to follow the first part of the Ethics. I think the concept that poses the greatest difficulty for me is that of attribute, and what it means for them to be infinite.

How I understand attributes

Here I explain the best guess I was able to come up with for what attributes are. It's probably wrong, as it leaves many of Spinoza's affirmations inexplicable, but I find it useful as a basis to ask more precise questions.

Let's look at the definition:

D4: By attribute I understand what the intellect perceives of a substance, as constituting its essence.

What the 'intellect' and 'essence' mean is not explained; from the way they are used I am guessing that the 'intellect' means something like 'logic and reason', and that 'essence' means 'definition' or 'identity'.

This is not so helpful. But there is another definition which mentions attributes:

D2: That thing is said to be finite in its own kind that can be limited by another of the same nature. For example, a body is called finite because we always conceive another that is greater. Thus a thought is limited by another thought. But a body is not limited by a thought nor a thought by a body.

The example he gives seem to indicate that 'in its own kind' refers to attributes. What he means by 'finite in its own kind' then can be rephrased as 'comparable, within an attribute.' So attributes are domains of the intellect within which things can be ordered.

Ordered according to what? Spinoza speaks explicitly of order in EIP33 and EIIP7, and in both cases it is clear he's talking about the order of cause and consequence. So my guess is that when he talks about a body 'that is greater' than another, he's not talking of its size (which wouldn't make sense for thoughts) but of its power to (be determined to) cause effects.

If that is the case, then an attribute would be a domain of the intellect which describes the power of things, i.e. which assigns causes and consequences. In other words it is simply a scientific or mathematical theory.

What confuses me about attributes

First question: What does 'an infinity of infinite attributes' mean?

In the definition of God Spinoza says:

I say absolutely infinite, not infinite in its own kind; for if something is only infinite in its own kind, we can conceive infinite attributes which do not pertain to its nature; but if something is absolutely infinite, whatever expresses essence and involves no negation pertains to its essence.

I think it would help if someone could give an example of something that is infinite in its own kind, and how one can conceive of an attribute which does not pertain to its nature.

Infinity here cannot be understood in the mathematical sense, where it can be approached by a never-ending sequence of numbers. By contrast, Spinoza's infinity seems beyond numbers altogether. His 'infinite' seems to mean something more like 'exhaustive'. A thing that is 'infinite' in this sense would leave nothing outside of it. But if one infinity already exhausts everything that there is, how can there be more than one infinity, a fortiori an infinity of them?

Also, it seems that in the phrase 'infinity of infinite attributes', there are two different kinds of infinities, the first of which is somehow less infinite than the second. The 'infinity of', as in 'infinity of attributes' seems to denote a set of simple elements; Spinoza is able to speak of 'infinite attributes, each of which expresses eternal and infinite essence' and to pick specific attributes like Thought and Extension. On the other hand, the 'infinity' of each 'infinite attributes' sounds more like a gunk. Indeed, if an attribute were ultimately made of atoms, the atoms would be understood through themselves, which is impossible since there is only one substance. But each attribute, as Spinoza shows in proposition 10, is conceived through itself, so it makes sense to collect them in a set. However this leads to my second question:

Second question: How are attributes distinguished?

According to proposition 10, 'Each attribute of a substance must be conceived through itself.' This should make them fall under the definition of substance. But Spinoza is very insistent that there is only one substance. So it should be the case that all attributes are identical and cannot be distinguished. Then how can it make sense to say that there is an infinity of them?

This is especially puzzling:

P9: The more reality or being each thing has, the more attributes belong to it.

Spinoza seems to think the definition of attributes requires no further justification. But what does it mean for a thing to have multiple essences? Does he mean equivalent essences? Wouldn't that just be different formulations of the same thing? But we can find as many equivalent definitions as we like for any thing just by inventing new languages, so everything should have the same degree of 'reality or being'. Is it only applicable to substance (since he talks specifically of 'attributes')? But there is only one substance, so what is the point of comparing degrees of reality?

There might be a way to explain this, which is to say that attributes are relative to an intellect. I don't think that's the case, but looking back at the definition ('what the intellect perceives of a substance'), I am a bit puzzled by the use of 'perceives' instead of something like 'asserts' or 'determines'. If I understand correctly, Spinoza's intellect consists of eternal truths, yet he uses a word which connotes a subjective point of view.

Another part which makes me think I might be missing something about the 'intellect' is in proposition 31, when he says 'By intellect (as is known through itself) we understand not absolute thought, but only a certain mode of thinking.' In my understanding, God's infinite intellect is identical with the world as the totality of facts. But if that is the case I don't see why the intellect is only a mode and not an attribute.

With my interpretation of 'attributes' as 'theories' it makes sense to understand attributes as relative to the power of an intellect. We need many theories to understand the world, because our finite intellect might perceive disconnected parts of the substance when there actually is a connection which we fail to notice. This explains how there can be many attributes when the substance is unique. But there's no way to make it fit the way Spinoza speaks of attributes. None of our theories can express the whole essence of substance, since they are finite. And our theories are not related through a one-to-one correspondance, as Spinoza claims of attributes.

Third question: If attributes are infinite, how can we tell that exactly two of them, not more or less, are accessible to the finite human intellect?

That's another issue with my interpretation: we have much more than two theories, in fact we can invent as many as we like. So among all those theories, how is it possible to recognise those which are part of different attributes? How do we know that theories of Thought (psychology, logic, linguistics...) and theories Extension (geometry, mechanics...) are not actually unified in God's infinite intellect? And how do we know that different theories of Extension (for instance mechanics and optics) are actually unified?

Fourth question: If all attributes are perceptions of the intellect, and if the intellect is a mode of thinking, doesn't it mean that all attributes are modes of thinking? How then can we know of anything outside of Thought?

Because it does seem to me that what Spinoza calls Extension is just a way of thinking we use to organise our ideas about motion. I don't see what can be known directly of Extension. For example, we know that sound is a vibration which affects our body in a particular way, which is associated with the idea our mind forms of that sound as it hits our eardrums. But the only thing we perceive directly is the idea. To get to know about sound as vibration, we had to be affected many vibrating things in many different ways, which allowed us find out something which is common to all of their ideas. We used this to build in our intellect a theory of vibration which explains, among other things, some of the properties of the sounds we hear. Which part of this happened outside of Thought?