Numbers are not things seen, heard, smelled, tasted, or touched. They are thoughts.

i still don't fully agree with this

if you'd extend that also to other describable qualities of things, like... roughness, sharpness, softness, etc., then sure

but when you feel two objects at once, you can tell based on the sensation that there are indeed two things there

twoness is a describable quality of matter--as is threeness, fourness, and so forth

not to undermine your argument, but i just don't think you can use that example

if you'd extend that also to other describable qualities of things, like... roughness, sharpness, softness, etc., then sure

but when you feel two objects at once, you can tell based on the sensation that there are indeed two things there

twoness is a describable quality of matter--as is threeness, fourness, and so forth

not to undermine your argument, but i just don't think you can use that example

The issue there is that softness, roughness, etc are indicators of touch. In context of Cognito Ergo Sum, the senses are wholly unreliable. You cannot use quality of matter as argument.

Quote from: eggsalad on February 13, 2016, 05:27:56 PM

Quote from: Prime Multivac on February 13, 2016, 05:26:24 PM

Quote from: eggsalad on February 13, 2016, 05:17:22 PM

Quote from: Prime Multivac on February 13, 2016, 05:01:51 PM

Quote from: eggsalad on February 13, 2016, 04:00:26 PM

Quote from: Prime Multivac on February 13, 2016, 12:21:28 AM

You cannot be tricked into thinking that you're thinking (because that means you're already thinking). Thinking inherently requires the capability of thought. If a being is thinks, then in can think; if it thinks, then it exists.

From what observation can you derive this if we have established that all observation we can make of this world is unreliable.

Thought it not perceived through the senses. In what way can one be manipulated into thinking without thinking in the first place? Thought, as the most basic rationality, cannot be deceived like touch, taste, smell, sight, and hearing.

if this is all a dream, you cannot rely on your five senses to seek knowledge. You can "see" green where green may not truly be; this is the basis of illusion. You cannot think where there is not the capability of thought because the existence of thought presumes thought. Thinking requires the ability to think.

What is thought without external stimuli? I can't envision thought ever being independent of external influences.

a + b = b + a, regardless of if we can see, hear, smell, taste, or touch. Thought is an entirely separate thing from the senses.

um how can you prove that without observation?

Well, for starters, it's an axiom. It's unquestionable.

Second, our concept of a number represented by a and a number represented by b is just that: conceptual. Numbers are not things seen, heard, smelled, tasted, or touched. They are thoughts.

What exactly gives axioms authority other than observed consistency? Are logical concepts not formed as a result of whether or not they are consistent within the context of our reality?

What exactly gives axioms authority other than observed consistency? Are logical concepts not formed as a result of whether or not they are consistent within the context of our reality?

this is where the conversation gets fuzzy, because what you're doing is questioning that which isn't supposed to be questioned

x = x (the reflexive property of equality) is axiomatic, because it functions axiomatically

if you question that property, you're not going to get a lucid answer--i can't tell you why the reflexive property of equality works; it just does, and i can demonstrate that to you

but as far as the hard reasoning behind it goes, there's literally no way to teach you that--it's just how reality works

so as far as i'm concerned, observed consistency (after accounting for all variables) is the only thing that you need in order to demonstrate an axiom

edit:
ninja'd

Quote from: eggsalad on February 13, 2016, 06:36:02 PM

What exactly gives axioms authority other than observed consistency? Are logical concepts not formed as a result of whether or not they are consistent within the context of our reality?

this is where the conversation gets fuzzy, because what you're doing is questioning that which isn't supposed to be questioned

x = x (the reflexive property of equality) is axiomatic, because it functions axiomatically

if you question that property, you're not going to get a lucid answer--i can't tell you why the reflexive property of equality works; it just does, and i can demonstrate that to you

but as far as the hard reasoning behind it goes, there's literally no way to teach you that--it's just how reality works

so as far as i'm concerned, observed consistency is the only thing that you need in order to demonstrate an axiom

So when we establish that, hypothetically, what we observe as reality is unreliable, what exactly exempts properties like these from scrutiny? That's the thing, what can be proven when you revoke the means of proving things? What differentiates between x = x or x =/= x when there are no avenues with which to verify it?

If there is no clear answer, then how can we say with certainty that if something thinks, it exists. Because that is a conclusion that follows as it would in observed reality, but in this scenario, that observed reality is under scrutiny.

I'm not sure why you keep using observation as a criterion, either; I should state again that the Dream Argument and Evil Demon Scenario presume that you cannot know anything by means of the senses. You can have observed consistency via logic, something that is not a means of the senses.

Aren't all thoughts abstractions made from observations of reality. Would a mind never exposed to our reality be able to formulate basic mathematical principles if it has never even been exposed to concepts such as plurality or oneness, if it has never observed a singular amount of something? Or observed two amounts and identify they are separate?

i'm not familiar with this but was just reading about it. might be relevant:

https://en.wikipedia.org/wiki/Kurt_G%C3%B6del#The_Incompleteness_Theorem

regarding the axiomatic system used to define arithmetic with natural numbers, "The consistency of the axioms cannot be proven within the system."

One of the reasons for axioms being unquestionable, I'd assume.

To Egg, I don't think this specific tangent is going anywhere. Despite what I may have said in the OP, Cogito isn't bulletproof by nay means. I think you're just looking at it a bit too closely. I'd point it out myself, but I'd be interested to see if you can find one of the flaws if you take a step back and look at the bigger picture.

But that sentence literally implies that they are questionable. "The axioms cannot be proven within the system"

After reading the page, it seems more related to the matter of not being able to compute an infinite log; for functional completion, one would require a complete set of all possibilities into a Boolean expression. If the data is incomplete, you cannot prove the expression for any variable. Therefore, if such an axiom were provable, it would be false. That's why, for any given axiomatic formula, there will be at least one true statement which cannot be proven.

This is why axioms are unquestionable; it's such a fuzzy piece of information that digging into the nitty-gritty requires a whole heaping of extra explanation on its own.

Quote from: eggsalad on February 13, 2016, 05:17:22 PM

Quote from: Prime Multivac on February 13, 2016, 05:01:51 PM

Quote from: eggsalad on February 13, 2016, 04:00:26 PM

Quote from: Prime Multivac on February 13, 2016, 12:21:28 AM

You cannot be tricked into thinking that you're thinking (because that means you're already thinking). Thinking inherently requires the capability of thought. If a being is thinks, then in can think; if it thinks, then it exists.

From what observation can you derive this if we have established that all observation we can make of this world is unreliable.

Thought it not perceived through the senses. In what way can one be manipulated into thinking without thinking in the first place? Thought, as the most basic rationality, cannot be deceived like touch, taste, smell, sight, and hearing.

if this is all a dream, you cannot rely on your five senses to seek knowledge. You can "see" green where green may not truly be; this is the basis of illusion. You cannot think where there is not the capability of thought because the existence of thought presumes thought. Thinking requires the ability to think.

What is thought without external stimuli? I can't envision thought ever being independent of external influences.

a + b = b + a, regardless of if we can see, hear, smell, taste, or touch. Thought is an entirely separate thing from the senses.

I'd like to chime in here and point out that you have to be more specific about what those variables represent, you can define an algebra with non commutative addition quite easily:

Consider a point on the equator of a sphere, if you move halfway around the equator (x), and then half way up from the equator to the north pole (y), you'll find yourself at the coordinate defined by  x + y
But if you move upwards first (y) then move the same distance parallel to the equator (x) then you end up a different point (y + x)
And if you refer to the picture below you should see why x + y ≠ y + x
Commutative addition isn't a universal property; you can create an algebra where any property does or does not hold provided it doesn't contradict itself.

Quote from: Prime Multivac on February 13, 2016, 05:26:24 PM

Quote from: eggsalad on February 13, 2016, 05:17:22 PM

Quote from: Prime Multivac on February 13, 2016, 05:01:51 PM

Quote from: eggsalad on February 13, 2016, 04:00:26 PM

Quote from: Prime Multivac on February 13, 2016, 12:21:28 AM

You cannot be tricked into thinking that you're thinking (because that means you're already thinking). Thinking inherently requires the capability of thought. If a being is thinks, then in can think; if it thinks, then it exists.

From what observation can you derive this if we have established that all observation we can make of this world is unreliable.

Thought it not perceived through the senses. In what way can one be manipulated into thinking without thinking in the first place? Thought, as the most basic rationality, cannot be deceived like touch, taste, smell, sight, and hearing.

if this is all a dream, you cannot rely on your five senses to seek knowledge. You can "see" green where green may not truly be; this is the basis of illusion. You cannot think where there is not the capability of thought because the existence of thought presumes thought. Thinking requires the ability to think.

What is thought without external stimuli? I can't envision thought ever being independent of external influences.

a + b = b + a, regardless of if we can see, hear, smell, taste, or touch. Thought is an entirely separate thing from the senses.

I'd like to chime in here and point out that you have to be more specific about what those variables represent, you can define an algebra with non commutative addition quite easily:

Consider a point on the equator of a sphere, if you move halfway around the equator (x), and then half way up from the equator to the north pole (y), you'll find yourself at the coordinate defined by  x + y
But if you move upwards first (y) then move the same distance parallel to the equator (x) then you end up a different point (y + x)
And if you refer to the picture below you should see why x + y ≠ y + x
Commutative addition isn't a universal property; you can create an algebra where any property does or does not hold provided it doesn't contradict itself.

but my example of the reflexive property was right on the money, right

Quote from: Cadenza on February 14, 2016, 05:34:42 PM

Quote from: Prime Multivac on February 13, 2016, 05:26:24 PM

Quote from: eggsalad on February 13, 2016, 05:17:22 PM

Quote from: Prime Multivac on February 13, 2016, 05:01:51 PM

Quote from: eggsalad on February 13, 2016, 04:00:26 PM

Quote from: Prime Multivac on February 13, 2016, 12:21:28 AM

You cannot be tricked into thinking that you're thinking (because that means you're already thinking). Thinking inherently requires the capability of thought. If a being is thinks, then in can think; if it thinks, then it exists.

From what observation can you derive this if we have established that all observation we can make of this world is unreliable.

Thought it not perceived through the senses. In what way can one be manipulated into thinking without thinking in the first place? Thought, as the most basic rationality, cannot be deceived like touch, taste, smell, sight, and hearing.

if this is all a dream, you cannot rely on your five senses to seek knowledge. You can "see" green where green may not truly be; this is the basis of illusion. You cannot think where there is not the capability of thought because the existence of thought presumes thought. Thinking requires the ability to think.

What is thought without external stimuli? I can't envision thought ever being independent of external influences.

a + b = b + a, regardless of if we can see, hear, smell, taste, or touch. Thought is an entirely separate thing from the senses.

I'd like to chime in here and point out that you have to be more specific about what those variables represent, you can define an algebra with non commutative addition quite easily:

Consider a point on the equator of a sphere, if you move halfway around the equator (x), and then half way up from the equator to the north pole (y), you'll find yourself at the coordinate defined by  x + y
But if you move upwards first (y) then move the same distance parallel to the equator (x) then you end up a different point (y + x)
And if you refer to the picture below you should see why x + y ≠ y + x
Commutative addition isn't a universal property; you can create an algebra where any property does or does not hold provided it doesn't contradict itself.

but my example of the reflexive property was right on the money, right

Yes, the equality relation is the clearest example of an equivalence relation; though what you said is bordering on tautology since an equivalence relation is defined to have reflexivity (along with transitivity and symmetry).(I would highly recommend this book for an overview, even if you only skim through it)

Like instead of trying to argue that equality must be reflexive since that's consistent with how we've been using it, it makes more sense to define a relation as being reflexive (and transitive and symmetric) and then simply call that relation "=", which is justified since it behaves exactly how you'd expect it to behave; if it behaved in any other way then it wouldn't be equivalence.

And the advantage of defining equality this way is that you can also define other equivalence relations from the same properties and instantly know how they behave even if they're dealing with something very different from numbers. For instance, Isomorphisms between vector spaces is pretty abstract on the surface, but since it obeys those three properties it behaves exactly the same as "=" even though it's not dealing with numbers and the vector spaces may be incredibly different from each-other in other ways.