Concept of Ganesa
- Shiva is process of perception and source of knowledge.
- It is the root desire of Brahma which started creation.
- In space, it is perception of hot and calm zones.
- These hot and calm zones can be differentiated or named into four parts according to their location from Sun.
The first part is called as Rudra which is within 100 diameters of sun. The second part is called Shiva which is between 100 to 1000 diameters of Sun. The third part is called as Shivatara which is between 1000 to 10,000 diameters of Sun and the last part is called as Shivatama which is between 104 to 105 diameters. Beyond that is Sadāshiva which is with Ambha. Ambha is water like spread of interstellar matter in galaxy, sun of it is Sāmba- Sadāśhiva. Within the human body, it is capability of perception.
As relates to the 3 pādas of Gāyatrī Mantra
Cause or source of perception is Shiva and its effect is Hanumān. Literally it means having 2 lips (Hanu). Junction of 2 lips is also the junction of karmendriya (organs of action) and Jñānendriya (organs of sense). So the person having complete harmony of action and wisdom is called Hanumān. First part of Gāyatrī mantra is creation process which happened after initial desire of Brahma. The creation was from ocean of Rasa like rain drops from cloud-called Vṛṣā (causing rains) and bull. Both are inputs of the basic yajña of farming. Creation was always like the previous one as rules are the same. Hence the creation process is also called Kapi which means monkey, as it has a habit of copying human actions. So Kapi also means to copy. Creator aspect is Vṛṣā-kapi, name of Hanumān in Vedas. Physical aspect indicated by part 2 of Gāyatrī mantra is motion caused by temperature difference, whose perception is Shiva. That process is called Māruti, i.e. born of Marut which means wind. Source and perception of knowledge is part 3 of Gāyatrī mantra. Its effect is Hanumān. Guru is Shiva, disciple or his improvement is Hanumān.
Chain of Guru-shiṣhya
This chain is indicated by Banyan tree which is symbol of Shiva. An aerial root descending from Banyan branch grows into a similar tree. Similarly, guru makes a man like himself by imparting knowledge to shiṣhya. Central banyan is Shiva, secondary trees are Hanumān. These trees are called Druma, and the complex of Druma is called Dumduma. There are Dumduma around places of Shiva. It is found around Lingarāja in Bhubaneshwar, around Dakṣhiṇeśhvara in Kolkata (Dumdum airport), near Kāmākhyā in Assam, Dumka near Baidyanath-dham in Jharkhand, Dumiyani near Somnath in Gujarat, and Damdami Taksal of Haramandir in Amritsar.
Shiva and it's aspect
Knowledge is an ordered structure. Any process creates more disorder, called entropy in second law of thermodynamics. If we acquire ordered knowledge, it creates more disorder in surroundings. By observing any object for knowing it, we disturb it. So, Shiva is Pralaya (deluge) aspect. Dance of creation is Rāsa (harmony) or Lāshya. Dance of destruction is Tāṇḍava (disorder, havoc) of Shiva. In human form, there were 28 incarnations of Shiva from Svāyambhuva Manu (Brahmā) in 29100 BC to Vedavyāsa in 3200 BC. They were sources of knowledge. In between many incarnations were crucial for destruction of asuras. There were 9 incarnations of Hanumān described in Parāśara Samhitā.
Knowledge and It's Structure
After acquiring, knowledge is structured in 2 ways, one structure is visible in which separate entities can be counted and other is abstract. Anything which can be counted (Gaṇanā) is Gaṇesha. Field of space or continuous matter is like Rasa (homogenous). Its knowledge is Sarasvatī (Sa + Rasa + vatī = That is with Rasa). These are opposite ends of knowledge unlike countable & abstract nouns, Bīja & Kṣetra Gaṇita (discrete, field theories) and Digital & Analog. After I explained this concept to Prof Abul Salam in 1979, he wrote an article or comment that controller of digital computer (Gaṇesha) should have his vehicle called mouse. Within the cobweb of 100 billion neurons of mind, our sensation moves as mouse moves on earth, so vehicle of Gaṇesha is mouse. Central part of human brain is like elephant head, called Gaṇesha guhā in tantra. Its trunk descends as Suṣumnā having 6 chakras till end of vertebral column.
Representation of knowledge in letters, words or sentences is countable and hence it is called Gaṇesha. Its meaning and sense is Sarasvatī, which cannot be counted. Both have been prayed in this manner by Tulsidas in his Hindi epic Rāma-charita-mānasa as belows :
These words indicate 3 types of infinity described in Cantor’s set, one which is countable with natural numbers (samkhyeya), second with rational numbers (asamkhyeya), and third is beyond expression by algebraic equations (Aparimeya). These names are also mentoned in Viṣhṇu-Sahasranāma-Ananta, asamkhyeya, aprameya. Ganesha is also collection of words (Pratyakṣha Brahma) or infinity of natural numbers. Its meaning is in between these discrete words Subrahma (Gopatha Brāhmaṇa, Pūrva 1/1), which means Brahma parts united by Sveda (water, sweating). This is like fuzzing of separate dots of ink sprayed by ink jet printer or pattern recognition by collection of dots which look separate on magnification or zooming. Physical form is called Subrahmaṇya (Kārttikeya), another son of human Shiva. Feelings in mind are Sarasvatī.
ॐ ब्रह्म ह वा इदमग्र आसीत्... तस्य श्रान्तस्य तप्तस्य सन्तप्तस्य ललाटे स्नेहो यदार्द्र्यमजायत। महद् वै यक्षं सुवेदं अविदामह इति। .. एतं सुवेदं सन्तं स्वेद इति आचक्षते॥१॥ सर्वेभ्यो रोमगर्त्तेभ्यः स्वेदधाराः प्रस्यन्दन्त। ... अहं इदं सर्वं धारयिष्यामि.... तस्मात् धारा अभवन्॥२॥
Brahma and his creation
In the beginning, there was Brahma only. Due to work and heat, sweat flowed from his forehead. That suveda was called sveda and it held (joined) roma-garttas (hair-roots, point objects). This way Brahma became Subrahma. The 3 types of infinities are also indicated in Īśāvāsya Upanishad-
स पर्यगात् शुक्रम् अकायम् अव्रणम् अस्नाविरम् शुद्धम् अपापविद्धम्। कविर्मनीषी परिभूः स्वयम्भूः याथातथ्यतोऽर्थान् व्यदधात् शाश्वतीभ्यः समाभ्यः। 
Abstract source is without form, defects, links etc. So by closing it in a boundary, forms links etc are created (in world and word also). The person who transforms the inner thoughts perfectly in words (or creates exact replica of abstract) makes eternal creations (poetry, or man etc as image of Brahma). Here, discrete objects are natural numbers, their links etc are rational numbers and the abstract source or field are transcendental numbers. Worship of Gaṇesha and Sarasvatī are at 2 ends of year cycle, separated by 6 months on Bhādra shukla 4 and Māgha shukla 5.
Invention of script
The first person deputed by Brahmā to create script was called Gaṇapati, who used ṛṇa (minus sign, dash) & chid-ṛṇa (small part of that, dot) to write letters. He used 3 pairs of dash-dot (like I-ching of China) to write symbols of Devas as letters. With 3 pairs, 26 to 64 letters are possible in Brāhmī script.
गणानां त्वा गणपतिं हवामहे, कविं कवीनामुपमश्रवस्तम्। ज्येष्ठराजं ब्रह्मणा ब्रह्मणस्पत आ नः शृण्वन्नृतिभिः सीद सादनम्॥१॥ विश्वेभ्यो हित्वा भुवनेभ्यस्परि त्वष्टाजनत् साम्नः कविः। स ऋणया चिदृणया ब्रह्मणस्पतिर्द्रुहो हन्तमह ऋतस्य धर्तरि॥१७॥
देवलक्ष्मं वै त्र्यालिखिता तामुत्तर लक्ष्माण देवा उपादधत....
Evolution of Karata from Ganesha
Population of a country is also countable and is called Gaṇa. Its head is called Gaṇapati or President. Servants of head of state steal despite any amount of vigilance appointed on them. That aspect is indicated by mouse. Sanskrit word Mūṣhik means thief. State runs by collection of tax from people. This action is indicated by elephant drawing water through its trunk. Hand of man is called ‘Kara in sanskrit; as it does work. For elephant, its trunk does all it's work, hence trunk is also called as ‘Kara’. Similar to drawing water by this ‘Kara’, tax is collected from people, thereby tax is also called as ‘Kara’. Gaṇesha also is praised as Karāṭa (collector of tax) in South India and especially Maharashtra.
तत् कराटाय च विद्द्महे हस्तिमुखाय धीमहि , तन्नो दन्ती प्रचोदयात् । 
There were many Gaṇapatis in human form called as Bṛhaspati, Angirā etc, who designed scripts in various eras.
Notes & References
- रामचरितमानस, मंगलाचरण
- गणपति अथर्व शीर्ष
- Ganapati Arharva-śīrṣa
- (गोपथ ब्राह्मण पूर्व १/१-२)
- (ईशावास्य उपनिषद् ८, वाज. यजु. ४०/८)
- (ऋक् २/२३/१, १७)
- (तैत्तिरीय संहिता ५/२/८/३)
- (कृष्णयजुर्वेदः - मैत्रायणीशाखा - अग्निचित्प्रकरणम् -119 - 120)