Does the Srimad Bhagavatam say the Earth is flat?
No, Srimad Bhagavatam doesn't say.
The reason I think the author suggests a flat earth is because of his background either from Christianity/Islam.
Trying to understand puranic consmology especially from Modern cosmology perspective is really challenging and not easy. The reason is the way puranas describe things is quite different.
Even Ayurveda, a medical branch deals in terms of jataragni, 3 doshas to analyse body etc.. Modern Medicine deals with body in different way. Trying to make sense of one from other view is not easy.
Modern science divides constituents of air in terms of argon, oxygen, nitrogen etc..Ancient methods divide air in terms of functions like apana, samana, prana, etc..
Actually both approaches are quite different and making sense of one from another is sometimes not possible.
I personally think it is better not to mix up both.
But, we can say that puranic cosmology had complex cosmological view not like just earth being flat etc.. as in bible or quran.
For eg: universes are described egg like in Brahma Samhita:
praty-aṇḍam evam ekāṁśād ekāṁśād viśati svayam sahasra-mūrdhā viśvātmā mahā-viṣṇuḥ sanātanaḥ
eko 'py asau racayituṁ jagad-aṇḍa-koṭiṁ yac-chaktir asti jagad-aṇḍa-cayā yad-antaḥ aṇḍāntara-stha-paramāṇu-cayāntara-stham- govindam ādi-puruṣaṁ tam ahaṁ bhajāmi
The living entities residing on Sumeru Mountain are always very warm, as at midday, because for them the sun is always overhead. Although the sun moves counterclockwise, facing the constellations, with Sumeru Mountain on its left, it also moves clockwise and appears to have the mountain on its right because it is influenced by the dakṣiṇāvarta wind. People living in countries at points diametrically opposite to where the sun is first seen rising will see the sun setting, and if a straight line were drawn from a point where the sun is at midday, the people in countries at the opposite end of the line would be experiencing midnight. Similarly, if people residing where the sun is setting were to go to countries diametrically opposite, they would not see the sun in the same condition.
This should give us idea how it was a complex model.
We don't have any ancient supplementary works explaining Puranic cosmology surviving and that makes it even more difficult to understand it.
The purpose of puranic cosmology is to understand the greatness of supreme lord.
Similarly there is Virat Rupa description in 2nd Canto, though Virat rupa is just a concept and thus imaginary form, it is described so that even a materialist can understand the greatness of supreme lord. Especially it was suggested that that form be meditated those who can't understand spiritual form of the Lord.
bhūrlokaḥ kalpitaḥ padbhyāṁ bhuvarloko ’sya nābhitaḥ svarlokaḥ kalpito mūrdhnā iti vā loka-kalpanā
Others may divide the whole planetary system into three divisions, namely the lower planetary systems on the legs [up to the earth], the middle planetary systems on the navel, and the upper planetary systems [Svarloka] from the chest to the head of the Supreme Personality.
SB 2.2.14: Unless the gross materialist develops a sense of loving service unto the Supreme Lord, the seer of both the transcendental and material worlds, he should remember or meditate upon the universal form of the Lord at the end of his prescribed duties.
SB 2.7.53: The Lord’s activities in association with His different energies should be described, appreciated and heard in accordance with the teachings of the Supreme Lord. If this is done regularly with devotion and respect, one is sure to get out of the illusory energy of the Lord.
SB 5.16.3: When the mind is fixed upon the Supreme Personality of Godhead in His external feature made of the material modes of nature — the gross universal form — it is brought to the platform of pure goodness. In that transcendental position, one can understand the Supreme Personality of Godhead, Vāsudeva, who in His subtler form is self-effulgent and beyond the modes of nature. O my lord, please describe vividly how that form, which covers the entire universe, is perceived.
Bhagavatam says these are 10 topics in it and just to described 10th item(Asraya Tattva, Sri Krishna), other items are described directly or indirectly in different ways.
SB 2.10.1: Śrī Śukadeva Gosvāmī said: In the Śrīmad-Bhāgavatam there are ten divisions of statements regarding the following: the creation of the universe, subcreation, planetary systems, protection by the Lord, the creative impetus, the change of Manus, the science of God, returning home, back to Godhead, liberation, and the summum bonum.
SB 2.10.2: To isolate the transcendence of the summum bonum, the symptoms of the rest are described sometimes by Vedic inference, sometimes by direct explanation, and sometimes by summary explanations given by the great sages.
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