How do time and space exist?

The best source for Mahayana is usually Nagarjuna. He proves that nothing really exists including time and space. He does not prove they do not exist but that it would not be correct to state positively that they do or do not exist. This is true for non-dualism generally. It is a doctrine of Unity so never endorses a distinction or division at a metaphysical level. All extreme views would have to be abandoned.

To understand why it would be incorrect to state that space-time either does or does not exist one would need to study Nagarjuna's Two Truths doctrine and his division of Reality into the Conventional and Ultimate realms or levels of description.

A doctrine of Unity requires that we never endorse an extreme or positive metaphysical theory, all of which Nagarjuna proves are logically indefensible. This is the reason why the language of mysticism has a contradictory and paradoxical appearance. For non-dualism true words seem paradoxical, as noted by Lao Tsu and explained by Nagarjuna. When Heraclitus says 'We are and are-not' he is also saying time and space are and are-not. This use of contradiction makes nondualism quite easy to spot where it appears.

Space-time would exist conventionally, as a component of our conventional world, but would be reducible thus not truly real. (It may be useful here to think about 'existence' and 'reality' as words with quite different meanings).

For a modern discussion of space-time that supports Nagarjuna's doctrine and the nondual view I'd recommend the mathematician Hermann Weyl. There's plenty of his writings online and somewhere an excellent article on his views by John Bell. He draws a distinction between time as we conceive of it in mathematics and everyday life, where it is merely a theory, and time as an intuitive or empirical continuum, which we do not experience and for which there is no evidence.

I feel it is not a good idea to ignore science, partly because the non-dual doctrine is in complete agreement with it, but scientists usually have little to say on these issues since space-time is a metaphysical explanandum. For a Buddhist discussion a recommended text would be Ven. Nyanaponika Thera's book Abhidhamma Studies: Buddhist Explorations of Consciousness and Time

I tried to link Mahayana emptiness to Theravada emptiness in this question.

Here's what I learnt about Mahayana emptiness:

According to Mahayana Madhyamaka emptiness (shunyata), all phenomena is empty of intrinsic essence (svabhava), and even this emptiness itself is empty of intrinsic essence. However, this intrinsic essence appears to be the essence given to phenomena by reification or objectification-classification (papanca). So this means that my mental idea of how some phenomena is, is not how it actually is.

Now, Nirvana is not a sankhara (conditioned and/or compounded thing) but it is also empty, in the sense that it is empty of the essence given to it by reification. So this means that my mental idea of how Nirvana is, is not how it actually is.

Even "emptiness is empty" means that my mental idea of how Mahayana emptiness is, is not how it actually is.

This is interesting, because it does not mean that a chair, a dog and Nirvana are mind-independently unreal or non-existent according to Madhyamaka. Rather, the mental idea that I have of a chair, a dog and Nirvana is unreal or non-existent.

And then about Theravada emptiness:

Now in Theravada, all suffering is related to clinging. Clinging is always related to the self. According to Sutta Nipata 4.14, the root of all reification or objectification-classification (papañca) is "I am the thinker".

Also from MN 1, an arahant who is fully liberated from suffering would see phenomena as they truly are, without reification where his mental idea of phenomena associates it with his self (of persons). This is apparently also known as tathata.

According to the Suñña Sutta, the five aggregates are empty of a self (of persons), including that they have no association with self (of persons). All reified mental ideas are mental fabrications (sankhara), so they too are empty of a self (of persons).

Now linking Theravada emptiness with Mahayana emptiness:

So, linking the Mahayana Madhyamaka emptiness to the Theravada emptiness, I can say that all phenomena is empty of a mentally reified intrinsic essence, where this reification or objectification-classification is rooted in "I am the thinker". Thanissaro Bhikkhu explains this as "the perception, 'I am the thinker' lies at the root of these classifications in that it reads into the immediate present a set of distinctions — I/not-I; being/not-being; thinker/thought; identity/non-identity — that then can proliferate into mental and physical conflict." I take it here that "I am the thinker" creates a duality between self (of persons) and non-self (of persons) that creates mental and physical conflict.

So, since Mahayana emptiness says that all phenomena is empty of mentally reified essence, and since all reification is rooted in a self (of persons), and Theravada emptiness states that all phenomena is empty of a self (of persons), then these two definitions of emptiness could be logically linked in this way.

Furthermore, the enlightened one who sees the Theravada emptiness of all phenomena through wisdom, will also simultaneously see the Mahayana emptiness of all phenomena, due to having his reification (papanca) ended, due to having his fetters concerning a self (of persons) uprooted. So, the enlightened one sees things as they truly are, which apparently is called tathata.

So, according to both Theravada and Mahayana, basically, Buddhism as a whole, it does not matter whether a chair or table or space or time or Nirvana or a cat or a tree are REALLY existing or not. All these things are definitely empty of the intrinsic essence given to it by mental reification.

There is more explanation in this answer.

For e.g. a dish of a meal consisting of meat may look like delicious food to a meat eater. At the same time it looks repulsive to a vegan. To a honey bee, it looks like a pile of dirt, because that's not its food.

So, the question is not whether delicious meal exists or repulsive food exists or dirt exists, in real life or just in your mind. That thing is simply empty of the essence given to it by the minds of a meat eater, a vegan and a honey bee.

In a way, you can say that the delicious meal only exists in the mind of the meat eater, and it does not exist outside. The repulsive food only exists in the mind of the vegan, and it does not exist outside. The piece of dirt only exists in the mind of the honey bee, and it does not exist outside. What is that which exists outside? Does it matter what it is? Does it matter how it exists?

Both Mahayana and Theravada are quite the same in my opinion. The only difference is that they focus on different perspectives of the same thing.

To the question, "How do time and space REALLY exist?", I answer, "not how you think it exists." That's the most accurate answer of Buddhism, from both Mahayana and Theravada, in my opinion.

The historical Buddha valued pragmatism over metaphysics. See the Parable of the Poisoned Arrow.

I think the Pali word for time is Kāla.

See also akalika. (the Dhamma is akalika i.e. "timeless").

A search for kala, kalika, and akalika on SuttaCentral doesn't return many entries.

It (the word akaliko) is mostly used in the context of the Dhammānussati.

The "Theravada" perspective presumably includes an Abhidhamma perspective, plus the "the post-canonical commentarial literature", in addition to what's in the suttas -- I don't know what these says about it, see perhaps (which I found using Google) Time and space: the Abhidhamma perspective:

Concept of time (Kala pannatti)

It is in the light of the distinction made between the dhammas as entities having objective reality on the one hand and pannattis as mental constructs, on the other, that we have to understand the Abhidhamma theory of time and space. On the subject of time the books of the Abhidhamma Pitaka are relatively silent. However, the fact that time is not elevated to the status of a conditioned or unconditioned dhamma shows that it has not been considered as existing in a real and ultimate sense. This is in contrast to the substantialist schools of Indian philosophy, where time is recognized as an eternal, all pervading substance, the existence of which is inferred from facts of consecution and simultaneity between phenomena. It is in the post-canonical commentarial literature that the Buddhist idea of time gets more clearly articulated.

The commentary to the Dhammasangani says that time is an avijjamana-pannatti, which means that it is a conceptual construct with no corresponding objective reality, a concept based on the continuous elemental flow. It is the dhammas, the ultimate constituents of empirical existence, which arise and perish in continual succession, that serve as a basis for our construction of the notion of time. only the dhammas are real (paramattha, saccikattha); time is a conceptual construct, a product of the interpretative function of our mind (kappana-siddha). Therefore, unlike the dhammas, time has no own-nature (sabhavato avijjamana).

That there is no time without reference to events and that, therefore, time is always determined by events, is aptly summarized by the following statement which we find expressed in the same commentary:

"Chronological time denoted by reference to this or that event is only a conventional expression"

etc.

I think Theravada Buddhism doesn't say anything about time or space-- I may be wrong

yes you're wrong, according to Buddhism space does exist, it is one of the six elements. But time is something made up by mind(according to my understanding).

And don't mix the ideas with Science, then you'll be in trouble to understand things.

In the original Pali scriptures, the term "dependently arisen" is used to refer to conditions related to the arising of "suffering" or "sorrow & grief". It is not used for any "causality" related to the laws of physics.

Dwelling at Savatthi... "Monks, I will describe & analyze dependent co-arising for you.

And what is dependent co-arising? From ignorance as a requisite condition come fabrications. From fabrications as a requisite condition comes consciousness. From consciousness as a requisite condition comes name-&-form. From name-&-form as a requisite condition come the six sense media. From the six sense media as a requisite condition comes contact. From contact as a requisite condition comes feeling. From feeling as a requisite condition comes craving. From craving as a requisite condition comes clinging/.. From clinging as a requisite condition comes becoming. From becoming as a requisite condition comes birth. From birth as a requisite condition, then aging & death, sorrow, lamentation, pain, distress, & despair come into play. Such is the origination of this entire mass of stress & suffering.

SN 12.2

As for the idea of "time", this can obviously relate to "craving" or "mental wanting" (such as thinking about the past & having expectations of the future, as described in MN 131). However this "mentally constructed time" (in MN 131) is obviously unrelated to the "time & space" of physics; as found in the following verse:

‘As the days and nights flit by, what sort of person am I becoming?’

AN 10.48

As for "space", it is one of the six elements (dhatu):

There are these six elements: the elements of earth, water, fire, air, space and consciousness.

When a mendicant knows and sees these six elements, they’re qualified to be called ‘skilled in the elements’.

MN 115

The reason a teaching about "space" is provided is so "space" is not clung to, as follows:

And what is the space element?

The space element may be interior or exterior.

And what is the interior space element?

Anything that’s space, spacious and organic that’s internal, pertaining to an individual. This includes: the ear canals, nostrils, and mouth; and the space for swallowing what is eaten and drunk, the space where it stays, and the space for excreting it from the nether regions. This is called the interior space element.

The interior space element and the exterior space element are just the space element. This should be truly seen with right understanding like this: ‘This is not mine, I am not this, this is not my self.’

When you truly see with right understanding, you reject the space element, detaching the mind from the space element.

MN 62

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Buddhism teaches about Pathavi,Apo.tejo,Vayo,Akasa (Sapace) and Vinnana (consciousness) Buddhism does not talk above the time even though it discusses the past, present and the future. The way I understand there is no absolute thing call time. Time is what we perceive when things change. Perhaps the closest Buddhist term for time is the word "Anicca" or impermanence.