All Facts Are Based in Faith

[Sunstone]

First of all, I didn't call anything "scientific faith,"....

I don't necessarily distinguish between "religious faith" and "mathematical faith" or "scientific faith"....

If you perhaps mean you did not endorse using the term "scientific faith", then I agree with you. But if you really do mean you "didn't call anything 'scientific faith'", then I think you are either giving the word "call" a peculiar meaning, or you are quite clearly confused about that. Choosing your terms with care will help to prevent miscommunication, and you should have the courtesy to define any terms you are using in unique ways.

...scientific facts are based on observation--which requires faith in the existence of a real world outside of our own heads, and faith in our perceptions and sensations to reflect that reality at least sometimes.

I do not believe it is true that scientific facts require us to have faith in the existence of an independent reality. So far as I can see, you have yet to demonstrate your thesis that scientific facts do require us to have faith in the existence of an independent reality. If that's so, please offer your reasons for believing that. On the other hand, if you have already laid out your reasons somewhere in this thread, please point me to where you have done so. As it is, I see you repeating your belief over and over, but I don't see where you've actually defended it.

Scientific facts are usually subject to replication and peer review, which require faith in the testimony of trusted others about their own personal experiences.

Could you elaborate on that, please? It's my own understanding that "peers" tend to be skeptical -- even extremely skeptical -- when reviewing each other's work. How does faith enter into the picture?

One must have faith in the reality of a world outside of their own head in order to determine facts from personal experience (observation).

Again, you keep repeating this claim, but you provide no argument for it so far as I've seen.

One must have faith in the testimony of trusted others to determine facts from the experiences of others (such as learning about the Leaning Tower of Pisa from books, or believing the testimony of the Gospels).

In this case, let's assume for the moment that what you said is true. Would that logically require us to believe that "all facts are based on faith"? Couldn't I establish by myself that X was a fact? And if I could, then not all facts are based on faith in the experiences of others.

I think your statement -- even if true -- would not require us to believe that all facts are based on faith.

...there is absolutely NOTHING you can know about the "real world" until you have faith that a "real world" exists outside of your own head, and that at least some of your perceptions and sensations are reflective of that reality.

I think your statement here is a bit muddled. I am guessing you meant once again to claim that "all facts are based on faith", or something to that effect. However, in my opinion your statement -- on close examination -- is vague and possibly ambiguous. If you were trying to say anything besides "all facts are based on faith", then please clarify your meaning. On the other hand, if that's all you meant, then please ignore this.

I know you enjoyed trotting out the big words...

I make a conscientious effort to express my points clearly. If you come across a term you are unfamiliar with, my advice is to google it or to ask how I am using the term.

...methodological naturalism just denies that any supernatural causes are possible explanations for physical phenomena.

Your statement about methodological naturalism strikes me as vague and possibly misguided. Here are some sources on the subject:

Wikipedia: Methodological Naturalism

The Basics of Philosophy: Types of Naturalism

Methodological Naturalism -- Boston College​

Methodological naturalism is the claim that there is no need to invoke the supernatural, including God, a god, or gods, in giving scientific explanations. The clear implication of no need to invoke the supernatural is that there is no need for any metaphysical assumptions at all -- very much including the metaphysical assumption that there exists independent of us an ontological reality.

In contrast, metaphysical naturalism is the claim that there is nothing supernatural, including God, a god, or gods. Metaphysical naturalism is loaded with loathsome metaphysical assumptions -- assumptions that are so close to Scientism they incestuously leg-hump it.

Methodological naturalism, unlike metaphysical naturalism, does not deny the existence of the supernatural, nor either deny or affirm any metaphysics, but merely denies the need to invoke such things in pursuing scientific inquiries or in arriving at scientific explanations and reliable, scientific facts.

 

[Jayhawker Soule]

"All Facts Are Based in Faith."

Well, at the very least, the above "Fact" is based in ill-founded faith. It's always fun when self-reference raises its perky head but, other than that, the OP is pretty underwhelming.

 

[Mock Turtle]

It seems like a rather artificial distinction to me. What "concrete evidence" do people have for belief in God? How about for faith in God?

Let's consult the book of faith statements about words.

belief: 1. an acceptance that a statement is true or that something exists. 2. trust, faith, or confidence in someone or something.

faith: 1. complete trust or confidence in someone or something. 2. strong belief in God or in the doctrines of a religion.

The two definitions are virtually parallel, right down to using each word to define the other.

Yeah, bit of a contradiction there. (from Collins)

Belief is a feeling of certainty that something exists, is true, or is good.
Your religious or political beliefs are your views on religious or political matters.
If it is your belief that something is the case, it is your strong opinion that it is the case.

If you have faith in someone or something, you feel confident about their ability or goodness.
A faith is a particular religion, for example Christianity, Buddhism, or Islam.
Faith is strong religious belief in a particular God.

Note the difference between believing your religious or political views are correct and your faith in such beliefs - not the same - one being a choice and the other a gamble on the quality of this belief.

 

[WalterTrull]

one being a choice and the other a gamble on the quality of this belief

Can't quite get my head around faith being a gamble. Wait a minute. OK, we're talking about different things. My bad.

 

[Mock Turtle]

Can't quite get my head around faith being a gamble. Wait a minute. OK, we're talking about different things. My bad.

That's how I see it - belief is just that whilst faith is the quality of that belief. Which to me makes them separate.

 

[Axe Elf]

@Polymath257 @Sunstone @Mostly Harmless too @Jayhawker Soule And anyone else I missed...

Wow, I really had no idea that such a simple epistemological principle would pose so much difficulty for so many! I do appreciate everyone who is participating in the discussion, but I can see that it's also time to reign in some of the peripheral topics that have come up and remind us all to stay on task here.

Don't worry, I will continue to respond to the subordinate minutiae as I have time to do so, but to get back to the heart of the thread, let's summarize the story so far.

The claim is made that all facts (things that we can know) are based in faith (the acceptance of propositions as being true in the absence of proof).

The three ways that we are able to know facts are:
1) By personal experience.
2) By the experience of others.
3) By the manipulation of symbols in formal systems of reasoning.

Each of these ways of identifying facts is based in faith:
1) Any knowledge from personal experience is based in the faith that a "real world" exists outside of our own heads, and that at least some of our perceptions and sensations are reflective of that reality.
2) Any knowledge from second-hand experience is based in the faith that the testimony of others having those experiences is reliable.
3) Any knowledge from formal systems of reasoning is based in the faith one has in the axioms underlying those formal systems.

So, to contradict the original claim that all facts are based in faith, one must do one of the following three things:
1) Demonstrate conclusively that the "real world" actually exists, and further, that there is a way to apprehend it directly, without the prerequisite faith in one's own nervous system to approximate the real world.
2) Demonstrate conclusively that we can know for sure when other people are testifying accurately about their own experiences and when they are not (without of course resorting to personal experience or the experience of others to resolve the issue, since that would be begging the question).
3) Demonstrate that the axioms of formal systems of reasoning can be proven to be true.

If anyone thinks they can do any of those three things, then go for it, but until then, understand that the claim itself remains unchallenged, despite all the various challenges to peripheral issues, word definitions, and ways of understanding the problem.

Ok, and now back to the wrangling of definitions and the torturing of reason...

 

[Axe Elf]

And does this sort of 'faith' based on evidence and testing differ in any substantial way from religious faith? The answer is clearly yes. So while the *confidence* one has in observation and testing is 'faith' in a very weak sense, it is nothing at all like religious faith that is not based on observation or testing.

First of all, I never said anything about ANY sort of 'faith' being based on evidence and testing. You somehow put the cart before the horse--evidence and testing is based on faith. Secondly, faith is faith. There are no different kinds. If you accept something as being true in the absence of proof, then you have faith in it. You can have faith in different things, but there are no different kinds of faith.

So I'm not sure what you are referencing when you say that "faith based on evidence and testing" (which doesn't exist anyway) "clearly" differs from religious faith.

 

[Axe Elf]

This seems to be an equivocation on the core term "meaning". @Polymath is talking about pragmatic or positivistic meaning, if I'm not mistaken, while you seem to be talking about emotional, religious, or spiritual meaning. Thus, it is no genuine criticism of Polymath's notion that Harman's brain in a vat concept is pragmatically meaningless that millions of people might find God spiritually meaningful despite that both BIV and God represent untestable concepts.

Meaning is meaning. Polymath denies that statements like "God exists" or "I am nothing more than a brain in a vat being stimulated by electrodes" have any meaning because they are untestable. This is not the case. The empirical evidence is clear that "God exists" has a great deal of meaning, at least for those who accept it as axiomatic. And it is clear that "I am nothing more than a brain in a vat being stimulated by electrodes" also has meaning, because we are able to discuss it.

As it happens, "you have to accept it on faith" does not necessarily follow from the fact "you can't test it". I can easily test whether hydrogen and oxygen combine to form water without assuming that hydrogen, oxygen, or water have any existence independent of my mind. That is, I can believe their independent status is unknowable.

Put differently, just as I can be an agnostic about the existence of God, I can be an agnostic about the existence of an independent reality.

Which is just to say that you cannot have any knowledge about an independent reality--that's what "agnostic" means (having no knowledge)--without accepting on faith that an independent reality exists, and that at least some of your perceptions and sensations are reflective of it. My claim is that all facts (knowledge) are based in faith, and this objection of yours is not related to knowledge but is based in agnosticism.

I disagree that axioms are really "statements assumed to be true".

You are allowed to disagree (to not have faith in the definition of axioms), but that is still what the term means. And it means what it means even if you argue that "meaning" is meaningless without meaning what you mean when you say "meaning" (or some such nonsense).

If we closely examine how axioms operate in logical and epistemological terms, then it would be more accurate to say that axioms function in much the same way as "please assume for the sake of discussion" statements. Or -- better yet, -- "prescriptive declarations". If one absolutely feels one must call axioms "statements assumed to be true", then one should at least add to that, "Axioms are largely analogous to statements assumed to be true."

This argument is largely analogous to arguments assumed to be nonsense.

"Please assume WHAT for the sake of discussion"? Oh yeah, please assume that something is TRUE for the sake of discussion.

 

[Axe Elf]

This is why I asked you for the definition of faith and specified that not all forms of faith are basis for fact, just as not all water supports fish life. It was a clarification.

To clarify further, there are no different "forms" of faith. Faith is faith--the acceptance of something as true in the absence of proof. There is misplaced faith--faith in something that is demonstrably false--and such faith is not suitable to support facts, just as polluted water is not suitable to support fish.

But that parallel lines don't intersect is self-evident since, if they did intersect, they would not be parallel. It's no different to saying there are no married bachelors. It's true by definition.

It's true by definition in Euclidean geometry but it is not true at all in other systems of geometry (which are no less useful in real life applications). In elliptical geometry, for instance two parallel lines intersect at least twice. Euclidean geometry is good for real-life applications to approximately flat surfaces. Elliptical geometry is good for real-life applications to curved surfaces, like a globe. Neither the axiom "two parallel lines never intersect" nor the axiom "two parallel lines intersect at least twice" are intrinsically more "true" than the other. One is perfectly free to assume that either one is true, depending on the kind of reasoning they want to do from their assumptions.

So yeah, there may be no married bachelors, but there ARE intersecting parallel lines.

 

[ImmortalFlame]

To clarify further, there are no different "forms" of faith. Faith is faith--the acceptance of something as true in the absence of proof. There is misplaced faith--faith in something that is demonstrably false--and such faith is not suitable to support facts, just as polluted water is not suitable to support fish.

You've just changed your definition. Earlier you said you were defining faith as "complete trust or confidence in someone or something", now you are saying it is "the acceptance of something as true in the absence of proof". Which definition are you using?

It's true by definition in Euclidean geometry but it is not true at all in other systems of geometry (which are no less useful in real life applications). In elliptical geometry, for instance two parallel lines intersect at least twice.

Please explain how this works, as parallel lines are necessarily defined as "two lines on a plane which do not meet or intersect".

Euclidean geometry is good for real-life applications to approximately flat surfaces. Elliptical geometry is good for real-life applications to curved surfaces, like a globe. Neither the axiom "two parallel lines never intersect" nor the axiom "two parallel lines intersect at least twice" are intrinsically more "true" than the other. One is perfectly free to assume that either one is true, depending on the kind of reasoning they want to do from their assumptions.

So yeah, there may be no married bachelors, but there ARE intersecting parallel lines.

How?

 

[Axe Elf]

You've just changed your definition. Earlier you said you were defining faith as "complete trust or confidence in someone or something", now you are saying it is "the acceptance of something as true in the absence of proof". Which definition are you using?

Either. Both. Take your pick. It doesn't matter. It only depends on what dictionary you have faith in.

Everyone who is not a dullard understands the word "faith."

Please explain how this works, as parallel lines are necessarily defined as "two lines which do not meet".

That's only in Euclidean geometry. In elliptic geometry, parallel lines necessarily meet at least twice.

How?

I thought I had explained it already, but I guess you need a better teacher.

Another way to describe the differences between these geometries is to consider two straight lines indefinitely extended in a two-dimensional plane that are both perpendicular to a third line:

• • In Euclidean geometry, the lines remain at a constant distance from each other (meaning that a line drawn perpendicular to one line at any point will intersect the other line and the length of the line segment joining the points of intersection remains constant) and are known as parallels.
  • In hyperbolic geometry, they "curve away" from each other, increasing in distance as one moves further from the points of intersection with the common perpendicular; these lines are often called ultraparallels.
  • In elliptic geometry, the lines "curve toward" each other and intersect.

Non-Euclidean geometry - Wikipedia

 

[ImmortalFlame]

Either. Both. Take your pick. It doesn't matter. It only depends on what dictionary you have faith in.

Everyone who is not a dullard understands the word "faith."

There's no need to be rude - I'm just trying to clarify exactly what definition of faith you mean as the word "faith" tends to be subject to a lot of equivocation between definitions.

That's only in Euclidean geometry. In elliptic geometry, parallel lines necessarily meet at least twice.

How?

I thought I had explained it already, but I guess you need a better teacher.

Another way to describe the differences between these geometries is to consider two straight lines indefinitely extended in a two-dimensional plane that are both perpendicular to a third line:

• 
  
  
  • In Euclidean geometry, the lines remain at a constant distance from each other (meaning that a line drawn perpendicular to one line at any point will intersect the other line and the length of the line segment joining the points of intersection remains constant) and are known as parallels.
  • In hyperbolic geometry, they "curve away" from each other, increasing in distance as one moves further from the points of intersection with the common perpendicular; these lines are often called ultraparallels.
  • In elliptic geometry, the lines "curve toward" each other and intersect.

Non-Euclidean geometry - Wikipedia

You do realize that what you've just quoted does not say that parallel lines intersect twice. It simply says that lines in Euclidean geometry that remain at a constant distance from each other are known as parallels. In other words, only when they are in a Euclidean dimension and remain at a CONSTANT DISTANCE AWAY from each other are they known as "parallels" It explicitly says "a line drawn perpendicular to one line at any point will intersect the other line" - meaning "a THIRD line drawn intersecting ONE of two parallel lines WILL INTERSECT THE OTHER", not that the parallel lines THEMSELVES will intersect, and it does not say that once the lines are moved into an elliptical space they REMAIN parallel:

"In geometry, parallel lines are lines in a plane which do not meet; that is, two lines in a plane that do not intersect or touch each other at any point are said to be parallel. By extension, a line and a plane, or two planes, in three-dimensional Euclidean space that do not share a point are said to be parallel. However, two lines in three-dimensional space which do not meet must be in a common plane to be considered parallel; otherwise they are called skew lines. Parallel planes are planes in the same three-dimensional space that never meet."
Parallel (geometry) - Wikipedia

 

[Axe Elf]

There's no need to be rude

I don't know that I'm being rude, but I AM being terse. The endless smokescreens of obfuscation are wearisome. We know what facts are, we know what faith is, we know that all facts are based in faith. It shouldn't be this difficult.

You do realize that what you've just quoted does not say that parallel lines intersect twice. It simply says that lines in Euclidean geometry that remain at a constant distance from each other are known as parallels. It explicitly says "a line drawn perpendicular to one line at any point will intersect the other line" - meaning "a THIRD line drawn intersecting ONE of two parallel lines WILL INTERSECT THE OTHER", not that the parallel lines THEMSELVES will intersect:

"In geometry, parallel lines are lines in a plane which do not meet; that is, two lines in a plane that do not intersect or touch each other at any point are said to be parallel. By extension, a line and a plane, or two planes, in three-dimensional Euclidean space that do not share a point are said to be parallel. However, two lines in three-dimensional space which do not meet must be in a common plane to be considered parallel; otherwise they are called skew lines. Parallel planes are planes in the same three-dimensional space that never meet."
Parallel (geometry) - Wikipedia

This is all just another smokescreen. No one is disputing that the definition of parallel lines in Euclidean geometry indicates that they never intersect, so proving that, as if you've actually accomplished something, is just attacking a straw man. No one is claiming that Euclid's parallel postulate means that parallel lines will intersect, and no one is mistaking the perpendicular for the parallel lines themselves, either. You've either completely misunderstood the evidence provided to you, or you are being intentionally obtuse.

If it is the former, go back and read again what I cited from Wikipedia (and this time pay close attention to the illustrations that accompany the text), particularly...

• 
  
  
  • In elliptic geometry, [parallel] lines "curve toward" each other and intersect.

And they intersect not just once, but at least twice.

 

[ImmortalFlame]

I don't know that I'm being rude, but I AM being terse. The endless smokescreens of obfuscation are wearisome. We know what facts are, we know what faith is, we know that all facts are based in faith. It shouldn't be this difficult.

I've not engaged in obfuscation. I'm actually trying to avoid that by nailing down the definitions you are using.

This is all just another smokescreen. No one is disputing that the definition of parallel lines in Euclidean geometry indicates that they never intersect, so proving that, as if you've actually accomplished something, is just attacking a straw man. No one is claiming that Euclid's parallel postulate means that parallel lines will intersect, and no one is mistaking the perpendicular for the parallel lines themselves, either. You've either completely misunderstood the evidence provided to you, or you are being intentionally obtuse.

If it is the former, go back and read again what I cited from Wikipedia (and this time pay close attention to the illustrations that accompany the text), particularly...

• 
  
  
  • In elliptic geometry, [parallel] lines "curve toward" each other and intersect.

And they intersect not just once, but at least twice.

Now you're being deliberately dishonest. The link says that the lines intersect, but it does not say that therefore they CONTINUE TO BE PARALLEL. In fact, it explicitly states that the lines are only defined as parallel when they REMAIN A CONSTANT DISTANCE FROM EACH OTHER IN EUCLIDEAN GEOMETRY. When they are moved into elliptical geometry, they intersect - and thus cease to be parallel

Read the page again.

 

[Axe Elf]

Now you're being deliberately dishonest. The link says that the lines intersect, but it does not say that therefore they CONTINUE TO BE PARALLEL. In fact, it explicitly states that the lines are only defined as parallel when they REMAIN A CONSTANT DISTANCE FROM EACH OTHER IN EUCLIDEAN GEOMETRY. When they are moved into elliptical geometry, they intersect - and thus cease to be parallel

Read the page again.

You're just making yourself look worse now. I'll give you a mercy pass.

 

[ImmortalFlame]

You're just making yourself look worse now. I'll give you a mercy pass.

Are you seriously going to pretend you didn't misread that wiki page?

 

[Jayhawker Soule]

I don't know that I'm being rude, but I AM being terse.

Now THAT was funny. But, contrary to its absurd (faith based) claim, facts meet the challenge of intersubjective viability.

 

[Polymath257]

Meaning is meaning. Polymath denies that statements like "God exists" or "I am nothing more than a brain in a vat being stimulated by electrodes" have any meaning because they are untestable. This is not the case. The empirical evidence is clear that "God exists" has a great deal of meaning, at least for those who accept it as axiomatic. And it is clear that "I am nothing more than a brain in a vat being stimulated by electrodes" also has meaning, because we are able to discuss it.

OK, discuss it. How do we determine whether we are brains in a vat? How is it possible to distinguish between being such and not? Or do you not agree that anything should be distinguishable from its negation?

The notion of God has emoitional meaning to a great many people, just like music and other art does. That isn't the same sort of meaning as when asking for truth value. Fiction is often quite emotionally meaningful,, even if not true.

Which is just to say that you cannot have any knowledge about an independent reality--that's what "agnostic" means (having no knowledge)--without accepting on faith that an independent reality exists, and that at least some of your perceptions and sensations are reflective of it. My claim is that all facts (knowledge) are based in faith, and this objection of yours is not related to knowledge but is based in agnosticism.

We understand your claim. We disagree. Even if we are brains in a vat, we can *still* look for patterns in our experiences, formulate hypotheses and test those hypotheses against future experiences. This is what allows us to *define* external reality as a means of organizing our hypotheses. This is NOT 'faith'.

You are allowed to disagree (to not have faith in the definition of axioms), but that is still what the term means. And it means what it means even if you argue that "meaning" is meaningless without meaning what you mean when you say "meaning" (or some such nonsense).

Sorry, but you are using technical words in an incorrect way. Formal systems use strings of symbols as their statements. There are rules of formation for well formed formulas. Axioms are initial formula from which deductions can occur using rules of inference, which are ways of going from one set of formulas to another. Definitions are abbreviations used to make communication more concise.

So, rules of inference are NOT axioms. Definitions are NOT axioms. At no stage is it assumed that any axiom is true. At no stage is it assumed that any rule of inference preserves truth value. Definitions are not assumptions. And NONE of these has to be taken on faith. But we can still do deductions in formal systems without taking any statement as having truth value or taking them on faith.

So, in math and logic, from a syntactical viewpoint (which is merely manipulation of symbols), there is no assumption made about whether the axioms are true 'in reality'. There is no assumption that any result applies to anything outside of that formal system.

Again, a perfect example of a formal system is the game of chess. It has a single 'axiom', which is the initial starting position. It has 'rules of inference' which are the possible moves of the pieces and the rule that players alternate moves. It has 'deductions' which are simply playing a game.

Math and logic are, technically, the same way.

Now, math and logic have been shown *useful* as a *language* for helping us formulate and test hypotheses about our experiences (see above). That utility has made some think that they are 'necessarily true'. But that is not the case. We use observation and testing to determine which mathematical and logical axioms are useful for our investigations.

Again, no *assumption* is made that math and logic are valid. No assumption is made about an external world. Instead, math and logic supply a language we use to formulate hypotheses about our experiences. One of those hyptheses that helps us organize our ideas is that there is an external world. Even if we are brains in a vat 'in reality', we would *still* make the assumption of an external world consistent with our experiences.

 

[Polymath257]

First of all, I never said anything about ANY sort of 'faith' being based on evidence and testing. You somehow put the cart before the horse--evidence and testing is based on faith. Secondly, faith is faith. There are no different kinds. If you accept something as being true in the absence of proof, then you have faith in it. You can have faith in different things, but there are no different kinds of faith.

So I'm not sure what you are referencing when you say that "faith based on evidence and testing" (which doesn't exist anyway) "clearly" differs from religious faith.

Well, then, you are missing the whole point. Science is based on observation and testing. So any notion is required to be testable to be meaningful. If a notion cannot be tested, it is literally not meaningful. At that point, we are talking nonsense. So, for example, the brains in a vat scenario, in the complete absence of evidence that we are brains in a vat or testability (no blue or red pill) is simply nonsense.

So, no, evidence and testing is NOT based on faith. Initially, we are all solipsists, but we use observation and testing to convince ourselves of the utility of logic and math and an that an external world is a more testable hypothesis than brains in a vat. That is NOT faith.

Religious faith, on the other hand, is *inherently* untestable (otherwise it would be science) and is thereby literally meaningless in the sense of truth value. It can still be *emotionally* meaningful, like literature or art, or any mythology. But it is not meaningful in the sense of having a truth value *because* it is untestable even in theory.

 

[Polymath257]

If it is the former, go back and read again what I cited from Wikipedia (and this time pay close attention to the illustrations that accompany the text), particularly...

• 
  
  
  • In elliptic geometry, [parallel] lines "curve toward" each other and intersect.

And they intersect not just once, but at least twice.

From the wikipedia article of parallel (geometry):

"In spherical geometry, all geodesics are great circles. Great circles divide the sphere in two equal hemispheres and all great circles intersect each other. Thus, there are no parallel geodesics to a given geodesic, as all geodesics intersect. Equidistant curves on the sphere are called parallels of latitude analogous to the latitude lines on a globe. Parallels of latitude can be generated by the intersection of the sphere with a plane parallel to a plane through the center of the sphere."

Notice the phrase: "there are no parallel geodesics to a given geodesic".

The definition of parallel has not changed: it still means the lines do not intersect. It is just that in elliptical geometry *there are no parallel lines* (where geodesics are the lines). So, it is NOT the case that parallel lines intersect in two points. That would be a contradiction to the definition and show the axioms are inconsistent. It is the case that there are just no parallel lines at all: every pair of lines intersects and is thereby NOT parallel.

There are also modifications made to the definition of parallel in the case of higher dimensional situations.